TSTP Solution File: NUM742^4 by E---3.1.00
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%------------------------------------------------------------------------------
% File : E---3.1.00
% Problem : NUM742^4 : TPTP v8.1.2. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_E %s %d THM
% Computer : n027.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat May 4 08:56:32 EDT 2024
% Result : Theorem 69.10s 9.23s
% Output : CNFRefutation 69.10s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats ran out of CPU time)
% Comments :
%------------------------------------------------------------------------------
thf(decl_22,type,
is_of: $i > ( $i > $o ) > $o ).
thf(decl_23,type,
all_of: ( $i > $o ) > ( $i > $o ) > $o ).
thf(decl_25,type,
in: $i > $i > $o ).
thf(decl_27,type,
emptyset: $i ).
thf(decl_42,type,
d_Sep: $i > ( $i > $o ) > $i ).
thf(decl_49,type,
omega: $i ).
thf(decl_58,type,
ap: $i > $i > $i ).
thf(decl_60,type,
d_Pi: $i > ( $i > $i ) > $i ).
thf(decl_61,type,
imp: $o > $o > $o ).
thf(decl_62,type,
d_not: $o > $o ).
thf(decl_67,type,
l_or: $o > $o > $o ).
thf(decl_123,type,
nat: $i ).
thf(decl_124,type,
n_is: $i > $i > $o ).
thf(decl_127,type,
n_some: ( $i > $o ) > $o ).
thf(decl_147,type,
diffprop: $i > $i > $i > $o ).
thf(decl_148,type,
d_29_ii: $i > $i > $o ).
thf(decl_149,type,
iii: $i > $i > $o ).
thf(decl_161,type,
times: $i > $i ).
thf(decl_162,type,
n_ts: $i > $i > $i ).
thf(decl_167,type,
d_1to: $i > $i ).
thf(decl_172,type,
n_2: $i ).
thf(decl_176,type,
pair1type: $i > $i ).
thf(decl_178,type,
first1: $i > $i > $i ).
thf(decl_179,type,
second1: $i > $i > $i ).
thf(decl_189,type,
frac: $i ).
thf(decl_191,type,
num: $i > $i ).
thf(decl_192,type,
den: $i > $i ).
thf(decl_193,type,
n_eq: $i > $i > $o ).
thf(decl_194,type,
moref: $i > $i > $o ).
thf(decl_195,type,
lessf: $i > $i > $o ).
thf(decl_197,type,
lesseq: $i > $i > $o ).
thf(decl_198,type,
epred1_2: $i > $i > $o ).
thf(decl_199,type,
epred2_2: $i > $i > $o ).
thf(decl_200,type,
epred3_2: $i > $i > $o ).
thf(decl_201,type,
epred4_2: $i > $i > $o ).
thf(decl_202,type,
epred5_2: $i > $i > $o ).
thf(decl_208,type,
esk1_0: $i ).
thf(decl_209,type,
esk2_0: $i ).
thf(decl_210,type,
esk3_0: $i ).
thf(def_nat,axiom,
( nat
= ( d_Sep @ omega
@ ^ [X1: $i] : ( X1 != emptyset ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_nat) ).
thf(def_l_or,axiom,
( l_or
= ( ^ [X83: $o] : ( imp @ ( d_not @ X83 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_l_or) ).
thf(def_d_not,axiom,
( d_not
= ( ^ [X76: $o] : ( imp @ X76 @ ~ $true ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_d_not) ).
thf(def_n_eq,axiom,
( n_eq
= ( ^ [X1: $i,X436: $i] : ( n_is @ ( n_ts @ ( num @ X1 ) @ ( den @ X436 ) ) @ ( n_ts @ ( num @ X436 ) @ ( den @ X1 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_n_eq) ).
thf(def_num,axiom,
( num
= ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_num) ).
thf(def_den,axiom,
( den
= ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_den) ).
thf(def_n_ts,axiom,
( n_ts
= ( ^ [X1: $i] : ( ap @ ( times @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_n_ts) ).
thf(def_lessf,axiom,
( lessf
= ( ^ [X1: $i,X450: $i] : ( iii @ ( n_ts @ ( num @ X1 ) @ ( den @ X450 ) ) @ ( n_ts @ ( num @ X450 ) @ ( den @ X1 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_lessf) ).
thf(def_iii,axiom,
( iii
= ( ^ [X1: $i,X211: $i] : ( n_some @ ( diffprop @ X211 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_iii) ).
thf(def_all_of,axiom,
( all_of
= ( ^ [X3: $i > $o,X2: $i > $o] :
! [X4: $i] :
( ( is_of @ X4 @ X3 )
=> ( X2 @ X4 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_all_of) ).
thf(def_is_of,axiom,
( is_of
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_is_of) ).
thf(def_moref,axiom,
( moref
= ( ^ [X1: $i,X449: $i] : ( d_29_ii @ ( n_ts @ ( num @ X1 ) @ ( den @ X449 ) ) @ ( n_ts @ ( num @ X449 ) @ ( den @ X1 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_moref) ).
thf(def_pair1type,axiom,
( pair1type
= ( ^ [X1: $i] :
( d_Pi @ ( d_1to @ n_2 )
@ ^ [X425: $i] : X1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_pair1type) ).
thf(def_lesseq,axiom,
( lesseq
= ( ^ [X1: $i,X466: $i] : ( l_or @ ( lessf @ X1 @ X466 ) @ ( n_eq @ X1 @ X466 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_lesseq) ).
thf(def_frac,axiom,
( frac
= ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',def_frac) ).
thf(satz41d,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X469: $i] : ( in @ X469 @ frac )
@ ^ [X470: $i] :
( ( lesseq @ X1 @ X470 )
=> ( d_not @ ( moref @ X1 @ X470 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz41d) ).
thf(satz37,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] : ( n_eq @ X1 @ X1 ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz37) ).
thf(satz51a,conjecture,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X493: $i] : ( in @ X493 @ frac )
@ ^ [X494: $i] :
( all_of
@ ^ [X4: $i] : ( in @ X4 @ frac )
@ ^ [X4: $i] :
( ( lesseq @ X1 @ X494 )
=> ( ( lessf @ X494 @ X4 )
=> ( lessf @ X1 @ X4 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz51a) ).
thf(satz41g,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X475: $i] : ( in @ X475 @ frac )
@ ^ [X476: $i] :
( ( moref @ X1 @ X476 )
=> ( d_not @ ( lesseq @ X1 @ X476 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz41g) ).
thf(satz41k,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X481: $i] : ( in @ X481 @ frac )
@ ^ [X482: $i] :
( ( d_not @ ( lesseq @ X1 @ X482 ) )
=> ( moref @ X1 @ X482 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz41k) ).
thf(satz50,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X491: $i] : ( in @ X491 @ frac )
@ ^ [X492: $i] :
( all_of
@ ^ [X4: $i] : ( in @ X4 @ frac )
@ ^ [X4: $i] :
( ( lessf @ X1 @ X492 )
=> ( ( lessf @ X492 @ X4 )
=> ( lessf @ X1 @ X4 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz50) ).
thf(satz42,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X457: $i] : ( in @ X457 @ frac )
@ ^ [X458: $i] :
( ( moref @ X1 @ X458 )
=> ( lessf @ X458 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz42) ).
thf(satz41e,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X471: $i] : ( in @ X471 @ frac )
@ ^ [X472: $i] :
( ( d_not @ ( moref @ X1 @ X472 ) )
=> ( lesseq @ X1 @ X472 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz41e) ).
thf(satz43,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X459: $i] : ( in @ X459 @ frac )
@ ^ [X460: $i] :
( ( lessf @ X1 @ X460 )
=> ( moref @ X460 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz43) ).
thf(satz44,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X461: $i] : ( in @ X461 @ frac )
@ ^ [X462: $i] :
( all_of
@ ^ [X4: $i] : ( in @ X4 @ frac )
@ ^ [X4: $i] :
( all_of
@ ^ [X13: $i] : ( in @ X13 @ frac )
@ ^ [X13: $i] :
( ( moref @ X1 @ X462 )
=> ( ( n_eq @ X1 @ X4 )
=> ( ( n_eq @ X462 @ X13 )
=> ( moref @ X4 @ X13 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz44) ).
thf(satz38,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X437: $i] : ( in @ X437 @ frac )
@ ^ [X438: $i] :
( ( n_eq @ X1 @ X438 )
=> ( n_eq @ X438 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz38) ).
thf(satz45,axiom,
( all_of
@ ^ [X1: $i] : ( in @ X1 @ frac )
@ ^ [X1: $i] :
( all_of
@ ^ [X463: $i] : ( in @ X463 @ frac )
@ ^ [X464: $i] :
( all_of
@ ^ [X4: $i] : ( in @ X4 @ frac )
@ ^ [X4: $i] :
( all_of
@ ^ [X13: $i] : ( in @ X13 @ frac )
@ ^ [X13: $i] :
( ( lessf @ X1 @ X464 )
=> ( ( n_eq @ X1 @ X4 )
=> ( ( n_eq @ X464 @ X13 )
=> ( lessf @ X4 @ X13 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p',satz45) ).
thf(c_0_27,plain,
( nat
= ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ),
inference(fof_simplification,[status(thm)],[def_nat]) ).
thf(c_0_28,plain,
( l_or
= ( ^ [Z0: $o] : ( imp @ ( imp @ Z0 @ ~ $true ) ) ) ),
inference(fof_simplification,[status(thm)],[def_l_or]) ).
thf(c_0_29,plain,
( d_not
= ( ^ [Z0: $o] : ( imp @ Z0 @ ~ $true ) ) ),
inference(fof_simplification,[status(thm)],[def_d_not]) ).
thf(c_0_30,plain,
( n_eq
= ( ^ [Z0: $i,Z1: $i] :
( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[def_n_eq]) ).
thf(c_0_31,axiom,
( num
= ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
inference(apply_def,[status(thm)],[def_num,c_0_27]) ).
thf(c_0_32,axiom,
( den
= ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) ) ) ),
inference(apply_def,[status(thm)],[def_den,c_0_27]) ).
thf(c_0_33,plain,
( n_ts
= ( ^ [Z0: $i] : ( ap @ ( times @ Z0 ) ) ) ),
inference(fof_simplification,[status(thm)],[def_n_ts]) ).
thf(c_0_34,plain,
( lessf
= ( ^ [Z0: $i,Z1: $i] :
( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) ) ) ) ) ),
inference(fof_simplification,[status(thm)],[def_lessf]) ).
thf(c_0_35,plain,
( iii
= ( ^ [Z0: $i,Z1: $i] : ( n_some @ ( diffprop @ Z1 @ Z0 ) ) ) ),
inference(fof_simplification,[status(thm)],[def_iii]) ).
thf(c_0_36,plain,
( all_of
= ( ^ [Z0: $i > $o,Z1: $i > $o] :
! [X4: $i] :
( ( Z0 @ X4 )
=> ( Z1 @ X4 ) ) ) ),
inference(fof_simplification,[status(thm)],[def_all_of]) ).
thf(c_0_37,plain,
( is_of
= ( ^ [Z0: $i,Z1: $i > $o] : ( Z1 @ Z0 ) ) ),
inference(fof_simplification,[status(thm)],[def_is_of]) ).
thf(c_0_38,plain,
( moref
= ( ^ [Z0: $i,Z1: $i] :
( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[def_moref]) ).
thf(c_0_39,plain,
( pair1type
= ( ^ [Z0: $i] :
( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z1: $i] : Z0 ) ) ),
inference(fof_simplification,[status(thm)],[def_pair1type]) ).
thf(c_0_40,plain,
( lesseq
= ( ^ [Z0: $i,Z1: $i] :
( imp
@ ( imp
@ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) ) ) )
@ ~ $true )
@ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) ) ) ) ),
inference(fof_simplification,[status(thm)],[def_lesseq]) ).
thf(c_0_41,plain,
( l_or
= ( ^ [Z0: $o] : ( imp @ ( imp @ Z0 @ ~ $true ) ) ) ),
inference(apply_def,[status(thm)],[c_0_28,c_0_29]) ).
thf(c_0_42,plain,
( n_eq
= ( ^ [Z0: $i,Z1: $i] :
( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[c_0_30,c_0_31]),c_0_32]),c_0_33]) ).
thf(c_0_43,plain,
( lessf
= ( ^ [Z0: $i,Z1: $i] :
( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[c_0_34,c_0_31]),c_0_32]),c_0_33]),c_0_35]) ).
thf(c_0_44,plain,
( all_of
= ( ^ [Z0: $i > $o,Z1: $i > $o] :
! [X4: $i] :
( ( Z0 @ X4 )
=> ( Z1 @ X4 ) ) ) ),
inference(apply_def,[status(thm)],[c_0_36,c_0_37]) ).
thf(c_0_45,plain,
( moref
= ( ^ [Z0: $i,Z1: $i] :
( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[c_0_38,c_0_31]),c_0_32]),c_0_33]) ).
thf(c_0_46,axiom,
( frac
= ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[def_frac,c_0_27]),c_0_39]) ).
thf(c_0_47,plain,
( lesseq
= ( ^ [Z0: $i,Z1: $i] :
( imp
@ ( imp
@ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) ) ) )
@ ~ $true )
@ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z2: $i] : ( Z2 != emptyset ) )
@ Z0 ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[c_0_40,c_0_41]),c_0_42]),c_0_43]) ).
thf(c_0_48,plain,
! [X512: $i] :
( ( in @ X512
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X511: $i] :
( ( in @ X511
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) ) )
| ( imp @ $false @ $false ) ) ) ) )
=> ( ( ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X511 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X512 ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) ) ) ) ),
inference(fool_unroll,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz41d]),c_0_44]),c_0_29]),c_0_45]),c_0_46]),c_0_47])]) ).
thf(c_0_49,plain,
! [X502: $i,X503: $i] :
( ( epred1_2 @ X503 @ X502 )
<=> ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) ) )
| ( imp @ $false @ $false ) ) ) ) ) ),
introduced(definition) ).
thf(c_0_50,plain,
! [X632: $i,X633: $i] :
( ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X633 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X632 ) ) )
| ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X633
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X632
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_48])])])])]) ).
thf(c_0_51,plain,
! [X502: $i,X503: $i] :
( ( epred1_2 @ X503 @ X502 )
=> ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) ) )
| ( imp @ $false @ $false ) ) ) ) ) ),
inference(split_equiv,[status(thm)],[c_0_49]) ).
thf(c_0_52,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ $true
| ~ $true
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_50]) ).
thf(c_0_53,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ $true
| ~ $true
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_50]) ).
thf(c_0_54,plain,
! [X600: $i] :
( ( in @ X600
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X600 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X600 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X600 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X600 ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz37]),c_0_44]),c_0_42]),c_0_46]) ).
thf(c_0_55,plain,
! [X749: $i,X750: $i] :
( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $true @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $true @ $false )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $true @ $true )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $true @ $false )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $false @ $false )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $false @ $true )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $false @ $false )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $false @ $true )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X750 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X749 ) ) ) )
| ~ ( epred1_2 @ X750 @ X749 ) ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_51])])])]) ).
thf(c_0_56,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_52]) ).
thf(c_0_57,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( imp @ ~ $true @ $true )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_53]) ).
thf(c_0_58,plain,
! [X727: $i] :
( ~ ( in @ X727
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X727 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X727 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X727 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X727 ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_54])])]) ).
thf(c_0_59,negated_conjecture,
~ ! [X503: $i] :
( ( in @ X503
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X502: $i] :
( ( in @ X502
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X501: $i] :
( ( in @ X501
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( epred1_2 @ X503 @ X502 )
=> ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X501 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X502 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X501 ) ) ) )
=> ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X501 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X503 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X501 ) ) ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(fool_unroll,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[inference(assume_negation,[status(cth)],[satz51a])]),c_0_44]),c_0_46]),c_0_47]),c_0_43])]),c_0_49]) ).
thf(c_0_60,plain,
! [X515: $i,X516: $i] :
( ( epred3_2 @ X516 @ X515 )
<=> ( ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ~ ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ~ ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) ) ),
introduced(definition) ).
thf(c_0_61,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_55]) ).
thf(c_0_62,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_55]) ).
thf(c_0_63,plain,
! [X4: $i,X1: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(cn,[status(thm)],[c_0_56]) ).
thf(c_0_64,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(cn,[status(thm)],[c_0_57]) ).
thf(c_0_65,plain,
! [X1: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_58]) ).
thf(c_0_66,negated_conjecture,
( ( in @ esk1_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
& ( in @ esk2_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
& ( in @ esk3_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
& ( epred1_2 @ esk1_0 @ esk2_0 )
& ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) )
& ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_59])])])]) ).
thf(c_0_67,plain,
! [X515: $i,X516: $i] :
( ( epred3_2 @ X516 @ X515 )
=> ( ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ~ ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ~ ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) ) ),
inference(split_equiv,[status(thm)],[c_0_60]) ).
thf(c_0_68,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_55]) ).
thf(c_0_69,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred1_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_61]) ).
thf(c_0_70,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(cn,[status(thm)],[c_0_62]) ).
thf(c_0_71,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(csr,[status(thm)],[c_0_63,c_0_64]) ).
thf(c_0_72,plain,
! [X1: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_65]) ).
thf(c_0_73,negated_conjecture,
( in @ esk3_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_66]) ).
thf(c_0_74,plain,
! [X753: $i,X754: $i] :
( ( $true
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $false )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $false )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $true )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $true )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $false )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $false )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $false )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $false )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $true )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $true )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $false )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $false )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $true )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $true )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $true @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ~ ( imp @ $false @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $false )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $false )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $true )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $true )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $false )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $false )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $false )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $false )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $true )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $true )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $false )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $false )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $true )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $true )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( $true
| ( imp @ $false @ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) )
& ( ~ $true
| ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X754 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X753 ) ) ) )
| ~ ( epred3_2 @ X754 @ X753 ) ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_67])])])]) ).
thf(c_0_75,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_55]) ).
thf(c_0_76,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ~ ( epred1_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_68]) ).
thf(c_0_77,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_69]) ).
thf(c_0_78,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_70]) ).
thf(c_0_79,plain,
! [X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(spm,[status(thm)],[c_0_71,c_0_72]) ).
thf(c_0_80,negated_conjecture,
( in @ esk3_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(cn,[status(thm)],[c_0_73]) ).
thf(c_0_81,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_82,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( imp @ ~ $true @ ~ $true )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_83,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_55]) ).
thf(c_0_84,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_85,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred1_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_75]) ).
thf(c_0_86,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_76]) ).
thf(c_0_87,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[c_0_77,c_0_78]) ).
thf(c_0_88,negated_conjecture,
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( imp @ ~ $true @ $true ) ),
inference(spm,[status(thm)],[c_0_79,c_0_80]) ).
thf(c_0_89,plain,
! [X517: $i,X518: $i] :
( ( epred4_2 @ X518 @ X517 )
<=> ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ~ ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ~ ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) ) ),
introduced(definition) ).
thf(c_0_90,plain,
! [X517: $i,X518: $i] :
( ( epred5_2 @ X518 @ X517 )
<=> ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) ),
introduced(definition) ).
thf(c_0_91,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_81]) ).
thf(c_0_92,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_82]) ).
thf(c_0_93,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred1_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_83]) ).
thf(c_0_94,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_95,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_84]) ).
thf(c_0_96,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_85]) ).
thf(c_0_97,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_86,c_0_87]),c_0_88]) ).
thf(c_0_98,plain,
! [X516: $i] :
( ( in @ X516
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X515: $i] :
( ( in @ X515
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X515 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X516 ) ) )
=> ( epred3_2 @ X516 @ X515 ) ) ) ),
inference(apply_def,[status(thm)],[inference(fool_unroll,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz41g]),c_0_44]),c_0_29]),c_0_45]),c_0_46]),c_0_47])]),c_0_60]) ).
thf(c_0_99,plain,
! [X518: $i] :
( ( in @ X518
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X517: $i] :
( ( in @ X517
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( ( epred4_2 @ X518 @ X517 )
& ( epred5_2 @ X518 @ X517 ) )
=> ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fool_unroll,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz41k]),c_0_44]),c_0_29]),c_0_45]),c_0_46]),c_0_47])]),c_0_89]),c_0_90]) ).
thf(c_0_100,plain,
! [X517: $i,X518: $i] :
( ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ~ ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ~ ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
=> ( epred4_2 @ X518 @ X517 ) ),
inference(split_equiv,[status(thm)],[c_0_89]) ).
thf(c_0_101,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(cn,[status(thm)],[c_0_91]) ).
thf(c_0_102,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(cn,[status(thm)],[c_0_92]) ).
thf(c_0_103,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_93]) ).
thf(c_0_104,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_94]) ).
thf(c_0_105,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_95]) ).
thf(c_0_106,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_86,c_0_96]),c_0_97]) ).
thf(c_0_107,plain,
! [X636: $i,X637: $i] :
( ~ ( in @ X636
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X637
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X636 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X637 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X637 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X636 ) ) )
| ( epred3_2 @ X636 @ X637 ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_98])])])]) ).
thf(c_0_108,plain,
! [X638: $i,X639: $i] :
( ~ ( in @ X638
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X639
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred4_2 @ X638 @ X639 )
| ~ ( epred5_2 @ X638 @ X639 )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X638 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X639 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X639 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X638 ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_99])])])]) ).
thf(c_0_109,plain,
! [X755: $i,X756: $i] :
( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X755 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X756 ) ) )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $true @ $false )
| ~ $true
| ( epred4_2 @ X756 @ X755 ) )
& ( $true
| ~ ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) )
& ( ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred4_2 @ X756 @ X755 ) ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_100])])])]) ).
thf(c_0_110,plain,
! [X513: $i,X514: $i] :
( ( epred2_2 @ X514 @ X513 )
<=> ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( imp @ $false @ $false ) ) ) ) ) ),
introduced(definition) ).
thf(c_0_111,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[c_0_101,c_0_102]) ).
thf(c_0_112,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[c_0_103,c_0_78]) ).
thf(c_0_113,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_104]) ).
thf(c_0_114,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_105,c_0_96]),c_0_88]),c_0_106]) ).
thf(c_0_115,plain,
! [X4: $i,X1: $i] :
( ( epred3_2 @ X1 @ X4 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_107]) ).
thf(c_0_116,plain,
! [X1: $i,X4: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred4_2 @ X1 @ X4 )
| ~ ( epred5_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_108]) ).
thf(c_0_117,plain,
! [X510: $i] :
( ( in @ X510
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X509: $i] :
( ( in @ X509
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X508: $i] :
( ( in @ X508
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X509 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X510 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X510 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X509 ) ) ) )
=> ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X508 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X509 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X509 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X508 ) ) ) )
=> ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X508 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X510 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X510 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X508 ) ) ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz50]),c_0_44]),c_0_46]),c_0_43]) ).
thf(c_0_118,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ ~ $true )
| ( imp @ $true @ $true )
| ( epred4_2 @ X4 @ X1 )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ $true ),
inference(split_conjunct,[status(thm)],[c_0_109]) ).
thf(c_0_119,plain,
! [X1: $i,X4: $i] :
( ( epred4_2 @ X1 @ X4 )
| ~ ( imp @ $true @ ~ $true )
| ~ $true ),
inference(split_conjunct,[status(thm)],[c_0_109]) ).
thf(c_0_120,plain,
! [X513: $i,X514: $i] :
( ( epred2_2 @ X514 @ X513 )
=> ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( imp @ $false @ $false ) ) ) ) ) ),
inference(split_equiv,[status(thm)],[c_0_110]) ).
thf(c_0_121,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_111,c_0_112]) ).
thf(c_0_122,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred1_2 @ X1 @ X4 )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_86,c_0_113]),c_0_97]),c_0_114]) ).
thf(c_0_123,plain,
! [X4: $i,X1: $i] :
( ( epred3_2 @ X1 @ X4 )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_115]) ).
thf(c_0_124,plain,
! [X1: $i,X4: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred4_2 @ X1 @ X4 )
| ~ ( epred5_2 @ X1 @ X4 ) ),
inference(cn,[status(thm)],[c_0_116]) ).
thf(c_0_125,negated_conjecture,
( in @ esk2_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_66]) ).
thf(c_0_126,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_55]) ).
thf(c_0_127,plain,
! [X629: $i,X630: $i,X631: $i] :
( ~ ( in @ X629
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X630
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X631
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X630 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X629 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X629 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X630 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X631 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X630 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X630 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X631 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X631 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X629 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X629 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X631 ) ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_117])])])]) ).
thf(c_0_128,plain,
! [X4: $i,X1: $i] :
( ( epred4_2 @ X4 @ X1 )
| ( imp @ $true @ $true )
| ( imp @ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_118]) ).
thf(c_0_129,plain,
! [X1: $i,X4: $i] :
( ( epred4_2 @ X1 @ X4 )
| ~ ( imp @ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_119]) ).
thf(c_0_130,plain,
! [X751: $i,X752: $i] :
( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $true @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $true @ $false )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $true @ $true )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $true @ $false )
| $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $false @ $false )
| $true
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $false @ $true )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $false @ $false )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $false @ $true )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X752 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X751 ) ) ) )
| ~ ( epred2_2 @ X752 @ X751 ) ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_120])])])]) ).
thf(c_0_131,plain,
! [X517: $i,X518: $i] :
( ( ( ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $true @ $true ) )
& ( ~ $true
| ~ ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ~ ( imp @ $false @ $true ) )
& ( ~ $true
| ~ ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ( imp @ $true @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ( imp @ $true @ $false ) ) ) )
& ( ( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ( imp @ $false @ $true ) )
& ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X517 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X518 ) ) )
| ( imp @ $false @ $false ) ) ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) )
=> ( epred5_2 @ X518 @ X517 ) ),
inference(split_equiv,[status(thm)],[c_0_90]) ).
thf(c_0_132,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_133,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ $true
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_134,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_135,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ $true
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_136,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ ( imp @ $true @ $true )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_105,c_0_113]),c_0_88]) ).
thf(c_0_137,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ $true @ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_114,c_0_121]) ).
thf(c_0_138,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ~ ( epred1_2 @ X1 @ X4 )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_122,c_0_121]) ).
thf(c_0_139,plain,
! [X1: $i,X4: $i] :
( ( epred3_2 @ X1 @ X4 )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred4_2 @ X1 @ X4 )
| ~ ( epred5_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_123,c_0_124]) ).
thf(c_0_140,negated_conjecture,
( in @ esk2_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(cn,[status(thm)],[c_0_125]) ).
thf(c_0_141,negated_conjecture,
( in @ esk1_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_66]) ).
thf(c_0_142,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred1_2 @ X1 @ X4 )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_126]) ).
thf(c_0_143,plain,
! [X1: $i,X4: $i,X5: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_127]) ).
thf(c_0_144,negated_conjecture,
( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_66]) ).
thf(c_0_145,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X4 @ X1 )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[c_0_128]),c_0_129]) ).
thf(c_0_146,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_130]) ).
thf(c_0_147,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_130]) ).
thf(c_0_148,plain,
! [X757: $i,X758: $i] :
( ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| $true
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ~ $true
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( imp @ $false @ $false )
| ( imp @ $false @ $true )
| ( imp @ $true @ $false )
| ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X757 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X758 ) ) )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( epred5_2 @ X758 @ X757 ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( epred5_2 @ X758 @ X757 ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( epred5_2 @ X758 @ X757 ) ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_131])])])]) ).
thf(c_0_149,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_132]) ).
thf(c_0_150,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_133]) ).
thf(c_0_151,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_134]) ).
thf(c_0_152,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(cn,[status(thm)],[c_0_135]) ).
thf(c_0_153,plain,
! [X615: $i] :
( ( in @ X615
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X614: $i] :
( ( in @ X614
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X615 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X614 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X614 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X615 ) ) )
=> ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X615 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X614 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X614 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X615 ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz42]),c_0_44]),c_0_45]),c_0_46]),c_0_43]) ).
thf(c_0_154,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_136,c_0_121]),c_0_137]),c_0_138]) ).
thf(c_0_155,negated_conjecture,
epred1_2 @ esk1_0 @ esk2_0,
inference(split_conjunct,[status(thm)],[c_0_66]) ).
thf(c_0_156,negated_conjecture,
! [X1: $i] :
( ( epred3_2 @ X1 @ esk2_0 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred4_2 @ X1 @ esk2_0 )
| ~ ( epred5_2 @ X1 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_139,c_0_140]) ).
thf(c_0_157,negated_conjecture,
( in @ esk1_0
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ),
inference(cn,[status(thm)],[c_0_141]) ).
thf(c_0_158,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(cn,[status(thm)],[c_0_142]) ).
thf(c_0_159,plain,
! [X1: $i,X5: $i,X4: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_143]) ).
thf(c_0_160,negated_conjecture,
( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ),
inference(cn,[status(thm)],[c_0_144]) ).
thf(c_0_161,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ $true )
| ( epred4_2 @ X4 @ X1 )
| ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
<~> ( imp @ $true @ $true ) )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(ext_eqfact,[status(thm)],[c_0_145]) ).
thf(c_0_162,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_130]) ).
thf(c_0_163,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred2_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_146]) ).
thf(c_0_164,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(cn,[status(thm)],[c_0_147]) ).
thf(c_0_165,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_130]) ).
thf(c_0_166,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ( epred5_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ $true ),
inference(split_conjunct,[status(thm)],[c_0_148]) ).
thf(c_0_167,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ $true ),
inference(split_conjunct,[status(thm)],[c_0_148]) ).
thf(c_0_168,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ( epred5_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ $true
| ~ $true ),
inference(split_conjunct,[status(thm)],[c_0_148]) ).
thf(c_0_169,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ $true @ $true ) ),
inference(split_conjunct,[status(thm)],[c_0_148]) ).
thf(c_0_170,plain,
! [X4: $i,X1: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( epred5_2 @ X4 @ X1 )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( imp @ $true @ ~ $true ) ),
inference(split_conjunct,[status(thm)],[c_0_148]) ).
thf(c_0_171,plain,
! [X514: $i] :
( ( in @ X514
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X513: $i] :
( ( in @ X513
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( ( ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( ( ~ ~ $true
| ( imp @ $true @ $true ) )
& ( ~ $true
| ( imp @ $true @ $false ) ) ) )
& ( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X513 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X514 ) ) )
| ( ( ~ ~ $true
| ( imp @ $false @ $true ) )
& ( ~ $true
| ( imp @ $false @ $false ) ) ) ) )
=> ( epred2_2 @ X514 @ X513 ) ) ) ),
inference(apply_def,[status(thm)],[inference(fool_unroll,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz41e]),c_0_44]),c_0_29]),c_0_45]),c_0_46]),c_0_47])]),c_0_110]) ).
thf(c_0_172,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ $true )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_74]) ).
thf(c_0_173,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_149]) ).
thf(c_0_174,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_150]) ).
thf(c_0_175,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_151]) ).
thf(c_0_176,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_152]) ).
thf(c_0_177,plain,
! [X741: $i,X742: $i] :
( ~ ( in @ X741
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X742
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X741 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X742 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X742 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X741 ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X741 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X742 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X742 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X741 ) ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_153])])])]) ).
thf(c_0_178,negated_conjecture,
( ( imp @ $true @ ~ $true )
| ~ ( epred3_2 @ esk1_0 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_154,c_0_155]) ).
thf(c_0_179,negated_conjecture,
( ( epred3_2 @ esk1_0 @ esk2_0 )
| ~ ( epred4_2 @ esk1_0 @ esk2_0 )
| ~ ( epred5_2 @ esk1_0 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_156,c_0_157]) ).
thf(c_0_180,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_158,c_0_78]),c_0_97]) ).
thf(c_0_181,negated_conjecture,
~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_66]) ).
thf(c_0_182,negated_conjecture,
! [X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_159,c_0_160]),c_0_80]),c_0_140])]) ).
thf(c_0_183,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X4 @ X1 )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[inference(cn,[status(thm)],[inference(dynamic_cnf,[status(thm)],[c_0_161])])]) ).
thf(c_0_184,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ~ ( epred2_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_162]) ).
thf(c_0_185,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( epred2_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_163]) ).
thf(c_0_186,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred2_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_164]) ).
thf(c_0_187,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ~ ( epred2_2 @ X1 @ X4 )
| ~ ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_165]) ).
thf(c_0_188,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X1 @ X4 )
| ( imp @ ~ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_166]) ).
thf(c_0_189,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_167]) ).
thf(c_0_190,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X1 @ X4 )
| ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_168]) ).
thf(c_0_191,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_169]) ).
thf(c_0_192,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X4 @ X1 )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_170]) ).
thf(c_0_193,plain,
! [X634: $i,X635: $i] :
( ( ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ $true
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| $true
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ $true
| $true
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ $true
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| $true
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| $true
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| $true
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X635 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X634 ) ) )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ $true
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( $true
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) )
& ( ~ ( imp @ $false @ $false )
| ~ ( imp @ $false @ $true )
| ~ ( imp @ $true @ $false )
| ~ ( imp @ $true @ $true )
| ( epred2_2 @ X634 @ X635 )
| ~ ( in @ X635
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X634
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_171])])])])]) ).
thf(c_0_194,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_172]) ).
thf(c_0_195,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[c_0_173,c_0_105]) ).
thf(c_0_196,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[c_0_174,c_0_175]),c_0_176]) ).
thf(c_0_197,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_177]) ).
thf(c_0_198,negated_conjecture,
( ( imp @ $true @ ~ $true )
| ~ ( epred4_2 @ esk1_0 @ esk2_0 )
| ~ ( epred5_2 @ esk1_0 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_178,c_0_179]) ).
thf(c_0_199,negated_conjecture,
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_180,c_0_155]) ).
thf(c_0_200,negated_conjecture,
~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) ),
inference(cn,[status(thm)],[c_0_181]) ).
thf(c_0_201,plain,
! [X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) )
| ( imp @ $true @ $true )
| ( epred4_2 @ X1 @ esk2_0 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_182,c_0_183]) ).
thf(c_0_202,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( epred2_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_184]) ).
thf(c_0_203,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred2_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[c_0_185,c_0_186]) ).
thf(c_0_204,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred2_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_187]) ).
thf(c_0_205,plain,
! [X1: $i,X4: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( epred5_2 @ X4 @ X1 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[c_0_188]),c_0_189]) ).
thf(c_0_206,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( epred5_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[c_0_190]),c_0_189]) ).
thf(c_0_207,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ $true ) ),
inference(cn,[status(thm)],[c_0_191]) ).
thf(c_0_208,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( epred5_2 @ X1 @ X4 )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( imp @ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_192]) ).
thf(c_0_209,plain,
! [X4: $i,X1: $i] :
( ( epred2_2 @ X1 @ X4 )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_193]) ).
thf(c_0_210,plain,
! [X617: $i] :
( ( in @ X617
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X616: $i] :
( ( in @ X616
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X616 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X617 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X617 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X616 ) ) ) )
=> ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X616 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X617 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X617 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X616 ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz43]),c_0_44]),c_0_45]),c_0_46]),c_0_43]) ).
thf(c_0_211,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred3_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_194]) ).
thf(c_0_212,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_195,c_0_196]) ).
thf(c_0_213,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_197]) ).
thf(c_0_214,plain,
( ( imp @ $true @ ~ $true )
| ~ ( epred4_2 @ esk1_0 @ esk2_0 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_198,c_0_189]),c_0_199]) ).
thf(c_0_215,negated_conjecture,
( ( epred4_2 @ esk1_0 @ esk2_0 )
| ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_200,c_0_201]),c_0_157])]) ).
thf(c_0_216,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_202,c_0_203]),c_0_88]) ).
thf(c_0_217,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_204,c_0_205]),c_0_206]),c_0_207]) ).
thf(c_0_218,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( epred5_2 @ X1 @ X4 )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_208,c_0_186]),c_0_199]),c_0_189]) ).
thf(c_0_219,plain,
! [X4: $i,X1: $i] :
( ( epred2_2 @ X1 @ X4 )
| ~ ( imp @ $true @ ~ $true )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_209]) ).
thf(c_0_220,plain,
! [X743: $i,X744: $i] :
( ~ ( in @ X743
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X744
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X744 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X743 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X743 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X744 ) ) ) )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X744 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X743 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X743 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X744 ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_210])])])]) ).
thf(c_0_221,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X4 @ X1 )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ $true
| ~ ( imp @ $true @ ~ $true )
| ~ ( imp @ $true @ $true ) ),
inference(split_conjunct,[status(thm)],[c_0_148]) ).
thf(c_0_222,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_211,c_0_175]),c_0_212]) ).
thf(c_0_223,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_204,c_0_213]) ).
thf(c_0_224,plain,
( ( imp @ $true @ $true )
| ( imp @ $true @ ~ $true ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_214,c_0_215]),c_0_199]) ).
thf(c_0_225,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_204,c_0_203]),c_0_216]) ).
thf(c_0_226,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ $true )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_217,c_0_218]) ).
thf(c_0_227,plain,
! [X4: $i,X1: $i] :
( ( epred2_2 @ X1 @ X4 )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_219]) ).
thf(c_0_228,plain,
! [X1: $i,X4: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_220]) ).
thf(c_0_229,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X4 @ X1 )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ) ),
inference(cn,[status(thm)],[c_0_221]) ).
thf(c_0_230,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( epred5_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_211,c_0_205]),c_0_222]),c_0_212]),c_0_189]) ).
thf(c_0_231,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( epred5_2 @ X1 @ X4 )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_208,c_0_176]),c_0_199]),c_0_189]) ).
thf(c_0_232,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred2_2 @ X1 @ X4 )
| ~ ( epred5_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_223,c_0_124]),c_0_129]),c_0_224]) ).
thf(c_0_233,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( epred5_2 @ X1 @ X4 )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_225,c_0_218]),c_0_226]),c_0_189]) ).
thf(c_0_234,plain,
! [X1: $i,X4: $i] :
( ( epred2_2 @ X1 @ X4 )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred5_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_227,c_0_124]),c_0_129]) ).
thf(c_0_235,plain,
! [X1: $i,X4: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_228]) ).
thf(c_0_236,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_229]) ).
thf(c_0_237,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( epred5_2 @ X1 @ X4 )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_230,c_0_231]) ).
thf(c_0_238,plain,
! [X1: $i] :
( ( imp @ $true @ $true )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred2_2 @ X1 @ X1 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_232,c_0_72]),c_0_233]) ).
thf(c_0_239,negated_conjecture,
! [X1: $i] :
( ( epred2_2 @ X1 @ esk3_0 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred5_2 @ X1 @ esk3_0 ) ),
inference(spm,[status(thm)],[c_0_234,c_0_80]) ).
thf(c_0_240,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
<~> ( imp @ ~ $true @ ~ $true ) )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(ext_eqfact,[status(thm)],[c_0_87]) ).
thf(c_0_241,plain,
! [X1: $i,X4: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( epred5_2 @ X4 @ X1 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(spm,[status(thm)],[c_0_235,c_0_205]) ).
thf(c_0_242,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ $true )
| ( epred5_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_236,c_0_175]),c_0_212]),c_0_230]),c_0_189]) ).
thf(c_0_243,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ $true )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_207,c_0_231]),c_0_237]) ).
thf(c_0_244,negated_conjecture,
( ( imp @ $true @ $true )
| ~ ( epred2_2 @ esk3_0 @ esk3_0 ) ),
inference(spm,[status(thm)],[c_0_238,c_0_80]) ).
thf(c_0_245,negated_conjecture,
( ( epred2_2 @ esk3_0 @ esk3_0 )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred5_2 @ esk3_0 @ esk3_0 ) ),
inference(spm,[status(thm)],[c_0_239,c_0_80]) ).
thf(c_0_246,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true )
| ~ ( epred1_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[inference(cn,[status(thm)],[inference(dynamic_cnf,[status(thm)],[c_0_240])])]) ).
thf(c_0_247,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_50]) ).
thf(c_0_248,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ( epred3_2 @ X4 @ X1 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(spm,[status(thm)],[c_0_123,c_0_241]) ).
thf(c_0_249,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_242,c_0_231]),c_0_243]) ).
thf(c_0_250,negated_conjecture,
( ( imp @ $true @ $true )
| ~ ( epred5_2 @ esk3_0 @ esk3_0 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_244,c_0_245]),c_0_224]) ).
thf(c_0_251,plain,
! [X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred1_2 @ X1 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_182,c_0_246]) ).
thf(c_0_252,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ $true )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_247]) ).
thf(c_0_253,plain,
! [X1: $i] :
( ( epred5_2 @ X1 @ X1 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ $true ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_248,c_0_72]),c_0_249]) ).
thf(c_0_254,plain,
( ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_250,c_0_189]) ).
thf(c_0_255,negated_conjecture,
( ( imp @ ~ $true @ $true )
| ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_200,c_0_251]),c_0_157]),c_0_155])]) ).
thf(c_0_256,plain,
! [X4: $i,X1: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ $true ) ),
inference(cn,[status(thm)],[c_0_252]) ).
thf(c_0_257,negated_conjecture,
( ( epred5_2 @ esk3_0 @ esk3_0 )
| ~ ( imp @ ~ $true @ $true ) ),
inference(spm,[status(thm)],[c_0_253,c_0_80]) ).
thf(c_0_258,negated_conjecture,
( ( imp @ ~ $true @ $true )
| ( imp @ $true @ $true ) ),
inference(spm,[status(thm)],[c_0_254,c_0_255]) ).
thf(c_0_259,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ $true
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_130]) ).
thf(c_0_260,plain,
! [X4: $i,X1: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( epred2_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ $true
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_193]) ).
thf(c_0_261,negated_conjecture,
~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_200,c_0_213]),c_0_157]),c_0_80])]) ).
thf(c_0_262,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( epred3_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ $true ) ),
inference(spm,[status(thm)],[c_0_123,c_0_256]) ).
thf(c_0_263,negated_conjecture,
imp @ $true @ $true,
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_250,c_0_257]),c_0_258]) ).
thf(c_0_264,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(split_conjunct,[status(thm)],[c_0_130]) ).
thf(c_0_265,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred2_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_259]) ).
thf(c_0_266,plain,
! [X4: $i,X1: $i] :
( ( epred2_2 @ X1 @ X4 )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_260]) ).
thf(c_0_267,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ $true
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ $true
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_50]) ).
thf(c_0_268,negated_conjecture,
( ~ ( epred4_2 @ esk3_0 @ esk1_0 )
| ~ ( epred5_2 @ esk3_0 @ esk1_0 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_261,c_0_124]),c_0_157]),c_0_80])]) ).
thf(c_0_269,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_236,c_0_213]) ).
thf(c_0_270,plain,
! [X4: $i,X1: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( epred3_2 @ X1 @ X4 )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_262,c_0_263])]),c_0_255]) ).
thf(c_0_271,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( epred5_2 @ X1 @ X4 )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_208,c_0_78]),c_0_199]),c_0_189]) ).
thf(c_0_272,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( epred4_2 @ X4 @ X1 )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ $true ),
inference(split_conjunct,[status(thm)],[c_0_109]) ).
thf(c_0_273,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( imp @ $true @ ~ $true )
| ( epred4_2 @ X1 @ X4 )
| ~ $true
| ~ $true ),
inference(split_conjunct,[status(thm)],[c_0_109]) ).
thf(c_0_274,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred2_2 @ X1 @ X4 )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_264]) ).
thf(c_0_275,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred2_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_265]) ).
thf(c_0_276,plain,
! [X4: $i,X1: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( epred2_2 @ X1 @ X4 )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_266]) ).
thf(c_0_277,plain,
! [X4: $i,X1: $i] :
( ( imp @ $true @ ~ $true )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_267]) ).
thf(c_0_278,plain,
( ~ ( epred4_2 @ esk3_0 @ esk1_0 )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_268,c_0_189]) ).
thf(c_0_279,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred5_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_269,c_0_124]),c_0_129]) ).
thf(c_0_280,plain,
! [X1: $i,X4: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred1_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_270,c_0_271]),c_0_249]),c_0_189]) ).
thf(c_0_281,plain,
! [X4: $i,X1: $i] :
( ( epred4_2 @ X4 @ X1 )
| ( imp @ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_272]) ).
thf(c_0_282,plain,
! [X4: $i,X1: $i] :
( ( epred4_2 @ X1 @ X4 )
| ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ ~ $true )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(cn,[status(thm)],[c_0_273]) ).
thf(c_0_283,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred2_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[c_0_274]) ).
thf(c_0_284,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_202,c_0_275]),c_0_216]) ).
thf(c_0_285,plain,
! [X4: $i,X1: $i] :
( ( epred2_2 @ X1 @ X4 )
| ( epred3_2 @ X1 @ X4 )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_123,c_0_276]) ).
thf(c_0_286,plain,
! [X1: $i,X4: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[c_0_277]) ).
thf(c_0_287,plain,
( ~ ( imp @ ~ $true @ ~ $true )
| ~ ( imp @ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_278,c_0_129]) ).
thf(c_0_288,negated_conjecture,
! [X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_182,c_0_213]),c_0_140])]) ).
thf(c_0_289,plain,
! [X4: $i,X1: $i] :
( ( epred5_2 @ X1 @ X4 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred5_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_279,c_0_263])]) ).
thf(c_0_290,negated_conjecture,
! [X1: $i] :
( ( epred5_2 @ X1 @ esk2_0 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred1_2 @ X1 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_280,c_0_140]) ).
thf(c_0_291,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( epred4_2 @ X4 @ X1 )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[c_0_281]),c_0_129]) ).
thf(c_0_292,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ ~ $true @ ~ $true )
| ( epred4_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[c_0_282]),c_0_129]) ).
thf(c_0_293,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( epred2_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[c_0_283,c_0_186]) ).
thf(c_0_294,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_105,c_0_275]),c_0_88]),c_0_284]) ).
thf(c_0_295,negated_conjecture,
! [X1: $i] :
( ( epred3_2 @ X1 @ esk2_0 )
| ( epred2_2 @ X1 @ esk2_0 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_285,c_0_140]) ).
thf(c_0_296,plain,
! [X1: $i,X4: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(csr,[status(thm)],[c_0_286,c_0_287]) ).
thf(c_0_297,negated_conjecture,
~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_200,c_0_288]),c_0_157])]) ).
thf(c_0_298,negated_conjecture,
! [X1: $i] :
( ( epred5_2 @ esk2_0 @ X1 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred5_2 @ X1 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_289,c_0_140]) ).
thf(c_0_299,negated_conjecture,
epred5_2 @ esk1_0 @ esk2_0,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_290,c_0_157]),c_0_155])]) ).
thf(c_0_300,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( epred4_2 @ X4 @ X1 ) ),
inference(csr,[status(thm)],[c_0_291,c_0_292]) ).
thf(c_0_301,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_111,c_0_293]) ).
thf(c_0_302,plain,
! [X1: $i,X4: $i] :
( ( imp @ ~ $true @ $true )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( epred2_2 @ X1 @ X4 )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_202,c_0_113]),c_0_216]),c_0_294]) ).
thf(c_0_303,negated_conjecture,
! [X1: $i] :
( ( epred2_2 @ X1 @ esk2_0 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred5_2 @ X1 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_234,c_0_140]) ).
thf(c_0_304,negated_conjecture,
( ( epred2_2 @ esk1_0 @ esk2_0 )
| ( epred3_2 @ esk1_0 @ esk2_0 )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(spm,[status(thm)],[c_0_295,c_0_157]) ).
thf(c_0_305,plain,
! [X621: $i] :
( ( in @ X621
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X620: $i] :
( ( in @ X620
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X619: $i] :
( ( in @ X619
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X618: $i] :
( ( in @ X618
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X621 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X620 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X620 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X621 ) ) )
=> ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X621 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X619 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X619 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X621 ) ) )
=> ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X620 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X618 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X618 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X620 ) ) )
=> ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X619 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X618 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X618 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X619 ) ) ) ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz44]),c_0_44]),c_0_45]),c_0_42]),c_0_46]) ).
thf(c_0_306,plain,
! [X602: $i] :
( ( in @ X602
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X601: $i] :
( ( in @ X601
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X602 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X601 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X601 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X602 ) ) )
=> ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X601 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X602 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X602 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X601 ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz38]),c_0_44]),c_0_42]),c_0_46]) ).
thf(c_0_307,plain,
! [X507: $i] :
( ( in @ X507
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X506: $i] :
( ( in @ X506
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X505: $i] :
( ( in @ X505
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ! [X504: $i] :
( ( in @ X504
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
=> ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X506 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X507 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X507 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X506 ) ) ) )
=> ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X507 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X505 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X505 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X507 ) ) )
=> ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X506 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X504 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X504 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X506 ) ) )
=> ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X504 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X505 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X505 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X504 ) ) ) ) ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[satz45]),c_0_44]),c_0_42]),c_0_46]),c_0_43]) ).
thf(c_0_308,negated_conjecture,
! [X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_182,c_0_296]),c_0_140])]) ).
thf(c_0_309,negated_conjecture,
( ~ ( epred4_2 @ esk2_0 @ esk1_0 )
| ~ ( epred5_2 @ esk2_0 @ esk1_0 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_297,c_0_124]),c_0_157]),c_0_140])]) ).
thf(c_0_310,negated_conjecture,
( ( epred5_2 @ esk2_0 @ esk1_0 )
| ~ ( imp @ $true @ ~ $true ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_298,c_0_157]),c_0_299])]) ).
thf(c_0_311,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( epred4_2 @ X4 @ X1 )
| ( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
<~> ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ) ),
inference(ext_eqfact,[status(thm)],[c_0_300]) ).
thf(c_0_312,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ $true @ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_294,c_0_301]) ).
thf(c_0_313,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ( imp @ ~ $true @ $true )
| ~ ( epred2_2 @ X1 @ X4 )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_302,c_0_301]) ).
thf(c_0_314,negated_conjecture,
( ( epred2_2 @ esk1_0 @ esk2_0 )
| ~ ( imp @ $true @ ~ $true )
| ~ ( epred5_2 @ esk1_0 @ esk2_0 ) ),
inference(spm,[status(thm)],[c_0_303,c_0_157]) ).
thf(c_0_315,negated_conjecture,
( ( epred2_2 @ esk1_0 @ esk2_0 )
| ( imp @ $true @ ~ $true ) ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_178,c_0_304]),c_0_199]) ).
thf(c_0_316,plain,
! [X745: $i,X746: $i,X747: $i,X748: $i] :
( ~ ( in @ X745
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X746
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X747
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X748
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X745 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X746 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X746 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X745 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X745 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X747 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X747 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X745 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X746 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X748 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X748 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X746 ) ) )
| ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X747 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X748 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X748 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X747 ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_305])])])]) ).
thf(c_0_317,plain,
! [X728: $i,X729: $i] :
( ~ ( in @ X728
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X729
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X728 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X729 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X729 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X728 ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X729 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X728 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X728 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X729 ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_306])])])]) ).
thf(c_0_318,plain,
! [X625: $i,X626: $i,X627: $i,X628: $i] :
( ~ ( in @ X625
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X626
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X627
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X628
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X626 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X625 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X625 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X626 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X625 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X627 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X627 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X625 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X626 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X628 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X628 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X626 ) ) )
| ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X628 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X627 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X627 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X628 ) ) ) ) ),
inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_307])])])]) ).
thf(c_0_319,negated_conjecture,
( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_200,c_0_308]),c_0_157])]) ).
thf(c_0_320,negated_conjecture,
~ ( imp @ $true @ ~ $true ),
inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_309,c_0_310]),c_0_129]) ).
thf(c_0_321,plain,
! [X4: $i,X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ( epred4_2 @ X4 @ X1 ) ),
inference(cn,[status(thm)],[inference(cn,[status(thm)],[inference(dynamic_cnf,[status(thm)],[c_0_311])])]) ).
thf(c_0_322,plain,
! [X1: $i,X4: $i] :
( ( imp @ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 )
| ~ ( epred2_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(csr,[status(thm)],[inference(spm,[status(thm)],[c_0_136,c_0_301]),c_0_312]),c_0_313]) ).
thf(c_0_323,negated_conjecture,
( ( epred3_2 @ esk1_0 @ esk2_0 )
| ~ ( epred4_2 @ esk1_0 @ esk2_0 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_179,c_0_299])]) ).
thf(c_0_324,negated_conjecture,
epred2_2 @ esk1_0 @ esk2_0,
inference(csr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_314,c_0_299])]),c_0_315]) ).
thf(c_0_325,plain,
! [X1: $i,X6: $i,X5: $i,X4: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X6
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_316]) ).
thf(c_0_326,plain,
! [X4: $i,X1: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_317]) ).
thf(c_0_327,plain,
! [X1: $i,X6: $i,X5: $i,X4: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X6
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) ),
inference(split_conjunct,[status(thm)],[c_0_318]) ).
thf(c_0_328,negated_conjecture,
( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_319,c_0_263])]) ).
thf(c_0_329,negated_conjecture,
imp @ ~ $true @ ~ $true,
inference(sr,[status(thm)],[c_0_199,c_0_320]) ).
thf(c_0_330,negated_conjecture,
! [X1: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ) )
| ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( epred4_2 @ X1 @ esk2_0 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(spm,[status(thm)],[c_0_182,c_0_321]) ).
thf(c_0_331,plain,
~ ( epred4_2 @ esk1_0 @ esk2_0 ),
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_322,c_0_323]),c_0_324])]),c_0_320]) ).
thf(c_0_332,plain,
! [X1: $i,X4: $i,X5: $i,X6: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X6
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_325]) ).
thf(c_0_333,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_326]) ).
thf(c_0_334,negated_conjecture,
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ( epred4_2 @ esk1_0 @ esk3_0 ) ),
inference(spm,[status(thm)],[c_0_200,c_0_300]) ).
thf(c_0_335,negated_conjecture,
( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_200,c_0_296]),c_0_80]),c_0_157])]) ).
thf(c_0_336,plain,
! [X1: $i,X4: $i] :
( ( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
<~> ( imp @ $true @ ~ $true ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(ext_eqfact,[status(thm)],[c_0_102]),c_0_287]) ).
thf(c_0_337,plain,
! [X1: $i,X4: $i,X5: $i,X6: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X6 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X6
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[c_0_327]) ).
thf(c_0_338,negated_conjecture,
! [X1: $i] :
( ( epred3_2 @ X1 @ esk3_0 )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred4_2 @ X1 @ esk3_0 )
| ~ ( epred5_2 @ X1 @ esk3_0 ) ),
inference(spm,[status(thm)],[c_0_139,c_0_80]) ).
thf(c_0_339,negated_conjecture,
( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_328,c_0_329])]) ).
thf(c_0_340,negated_conjecture,
( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) ),
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_200,c_0_330]),c_0_157])]),c_0_331]) ).
thf(c_0_341,plain,
! [X1: $i,X4: $i,X5: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X5 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X5
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(spm,[status(thm)],[c_0_332,c_0_72]) ).
thf(c_0_342,negated_conjecture,
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
| ( epred4_2 @ esk1_0 @ esk3_0 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_333,c_0_334]),c_0_80]),c_0_157])]) ).
thf(c_0_343,negated_conjecture,
( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( imp @ $true @ $true )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred4_2 @ esk1_0 @ esk3_0 ) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_335,c_0_124]),c_0_80]),c_0_157])]),c_0_189]) ).
thf(c_0_344,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(csr,[status(thm)],[inference(cn,[status(thm)],[inference(cn,[status(thm)],[inference(dynamic_cnf,[status(thm)],[c_0_336])])]),c_0_199]) ).
thf(c_0_345,negated_conjecture,
! [X1: $i,X4: $i] :
( ( n_some
@ ( diffprop
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_337,c_0_160]),c_0_80]),c_0_140])]) ).
thf(c_0_346,negated_conjecture,
( ( epred3_2 @ esk1_0 @ esk3_0 )
| ~ ( epred4_2 @ esk1_0 @ esk3_0 )
| ~ ( epred5_2 @ esk1_0 @ esk3_0 ) ),
inference(spm,[status(thm)],[c_0_338,c_0_157]) ).
thf(c_0_347,plain,
! [X1: $i,X4: $i] : ( epred5_2 @ X1 @ X4 ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_189,c_0_329])]) ).
thf(c_0_348,negated_conjecture,
~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_339,c_0_340])]) ).
thf(c_0_349,negated_conjecture,
! [X1: $i] :
( ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ( epred4_2 @ esk1_0 @ esk3_0 )
| ~ ( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_341,c_0_342]),c_0_157]),c_0_80])]) ).
thf(c_0_350,negated_conjecture,
( d_29_ii
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_235,c_0_160]),c_0_80]),c_0_140])]) ).
thf(c_0_351,negated_conjecture,
( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( imp @ ~ $true @ ~ $true )
| ~ ( epred4_2 @ esk1_0 @ esk3_0 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_343,c_0_263])]) ).
thf(c_0_352,plain,
! [X1: $i,X4: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ( imp @ $true @ ~ $true )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X4 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
| ~ ( in @ X4
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred3_2 @ X1 @ X4 ) ),
inference(spm,[status(thm)],[c_0_344,c_0_345]) ).
thf(c_0_353,negated_conjecture,
( ( epred3_2 @ esk1_0 @ esk3_0 )
| ~ ( epred4_2 @ esk1_0 @ esk3_0 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_346,c_0_347])]) ).
thf(c_0_354,negated_conjecture,
epred4_2 @ esk1_0 @ esk3_0,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_348,c_0_349]),c_0_350]),c_0_140])]) ).
thf(c_0_355,negated_conjecture,
( ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) )
| ~ ( epred4_2 @ esk1_0 @ esk3_0 ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_351,c_0_329])]) ).
thf(c_0_356,plain,
! [X1: $i] :
( ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) ) )
| ~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ X1 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) )
| ~ ( in @ X1
@ ( d_Pi @ ( d_1to @ n_2 )
@ ^ [Z0: $i] :
( d_Sep @ omega
@ ^ [Z1: $i] : ( Z1 != emptyset ) ) ) )
| ~ ( epred3_2 @ X1 @ esk3_0 ) ),
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_352,c_0_72]),c_0_80])]),c_0_320]) ).
thf(c_0_357,negated_conjecture,
( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk2_0 ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_333,c_0_340]),c_0_140]),c_0_157])]) ).
thf(c_0_358,negated_conjecture,
epred3_2 @ esk1_0 @ esk3_0,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_353,c_0_354])]) ).
thf(c_0_359,negated_conjecture,
~ ( n_is
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( ap
@ ( times
@ ( first1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk3_0 ) )
@ ( second1
@ ( d_Sep @ omega
@ ^ [Z0: $i] : ( Z0 != emptyset ) )
@ esk1_0 ) ) ),
inference(cn,[status(thm)],[inference(rw,[status(thm)],[c_0_355,c_0_354])]) ).
thf(c_0_360,plain,
$false,
inference(sr,[status(thm)],[inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_356,c_0_357]),c_0_157]),c_0_358])]),c_0_359]),
[proof] ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.15 % Problem : NUM742^4 : TPTP v8.1.2. Released v7.1.0.
% 0.10/0.16 % Command : run_E %s %d THM
% 0.16/0.36 % Computer : n027.cluster.edu
% 0.16/0.36 % Model : x86_64 x86_64
% 0.16/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.36 % Memory : 8042.1875MB
% 0.16/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.36 % CPULimit : 300
% 0.16/0.36 % WCLimit : 300
% 0.16/0.36 % DateTime : Fri May 3 09:58:20 EDT 2024
% 0.16/0.36 % CPUTime :
% 0.21/0.52 Running higher-order theorem proving
% 0.21/0.52 Running: /export/starexec/sandbox2/solver/bin/eprover-ho --delete-bad-limit=2000000000 --definitional-cnf=24 -s --print-statistics -R --print-version --proof-object --auto-schedule=8 --cpu-limit=300 /export/starexec/sandbox2/tmp/tmp.nolHxfieC5/E---3.1_17910.p
% 69.10/9.23 # Version: 3.1.0-ho
% 69.10/9.23 # Preprocessing class: HSLMSMSMLLLCHSA.
% 69.10/9.23 # Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 69.10/9.23 # Starting pre_casc_4 with 1200s (4) cores
% 69.10/9.23 # Starting full_lambda_6 with 300s (1) cores
% 69.10/9.23 # Starting sh10 with 300s (1) cores
% 69.10/9.23 # Starting post_as_ho9 with 300s (1) cores
% 69.10/9.23 # Starting post_as_ho8 with 300s (1) cores
% 69.10/9.23 # post_as_ho9 with pid 17991 completed with status 0
% 69.10/9.23 # Result found by post_as_ho9
% 69.10/9.23 # Preprocessing class: HSLMSMSMLLLCHSA.
% 69.10/9.23 # Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 69.10/9.23 # Starting pre_casc_4 with 1200s (4) cores
% 69.10/9.23 # Starting full_lambda_6 with 300s (1) cores
% 69.10/9.23 # Starting sh10 with 300s (1) cores
% 69.10/9.23 # Starting post_as_ho9 with 300s (1) cores
% 69.10/9.23 # SinE strategy is GSinE(CountFormulas,,true,1,0,2,20000,1.0,true)
% 69.10/9.23 # Search class: HGUNF-FSLM21-DHSMMFBN
% 69.10/9.23 # partial match(5): HGUSM-FSLM31-DHSFFFBN
% 69.10/9.23 # Scheduled 6 strats onto 1 cores with 300 seconds (300 total)
% 69.10/9.23 # Starting new_bool_5 with 135s (1) cores
% 69.10/9.23 # new_bool_5 with pid 18001 completed with status 0
% 69.10/9.23 # Result found by new_bool_5
% 69.10/9.23 # Preprocessing class: HSLMSMSMLLLCHSA.
% 69.10/9.23 # Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 69.10/9.23 # Starting pre_casc_4 with 1200s (4) cores
% 69.10/9.23 # Starting full_lambda_6 with 300s (1) cores
% 69.10/9.23 # Starting sh10 with 300s (1) cores
% 69.10/9.23 # Starting post_as_ho9 with 300s (1) cores
% 69.10/9.23 # SinE strategy is GSinE(CountFormulas,,true,1,0,2,20000,1.0,true)
% 69.10/9.23 # Search class: HGUNF-FSLM21-DHSMMFBN
% 69.10/9.23 # partial match(5): HGUSM-FSLM31-DHSFFFBN
% 69.10/9.23 # Scheduled 6 strats onto 1 cores with 300 seconds (300 total)
% 69.10/9.23 # Starting new_bool_5 with 135s (1) cores
% 69.10/9.23 # Preprocessing time : 0.015 s
% 69.10/9.23 # Presaturation interreduction done
% 69.10/9.23
% 69.10/9.23 # Proof found!
% 69.10/9.23 # SZS status Theorem
% 69.10/9.23 # SZS output start CNFRefutation
% See solution above
% 69.10/9.25 # Parsed axioms : 565
% 69.10/9.25 # Removed by relevancy pruning/SinE : 499
% 69.10/9.25 # Initial clauses : 11240
% 69.10/9.25 # Removed in clause preprocessing : 10869
% 69.10/9.25 # Initial clauses in saturation : 371
% 69.10/9.25 # Processed clauses : 6962
% 69.10/9.25 # ...of these trivial : 22
% 69.10/9.25 # ...subsumed : 4049
% 69.10/9.25 # ...remaining for further processing : 2891
% 69.10/9.25 # Other redundant clauses eliminated : 0
% 69.10/9.25 # Clauses deleted for lack of memory : 0
% 69.10/9.25 # Backward-subsumed : 988
% 69.10/9.25 # Backward-rewritten : 646
% 69.10/9.25 # Generated clauses : 48137
% 69.10/9.25 # ...of the previous two non-redundant : 27827
% 69.10/9.25 # ...aggressively subsumed : 0
% 69.10/9.25 # Contextual simplify-reflections : 669
% 69.10/9.25 # Paramodulations : 33535
% 69.10/9.25 # Factorizations : 0
% 69.10/9.25 # NegExts : 0
% 69.10/9.25 # Equation resolutions : 0
% 69.10/9.25 # Disequality decompositions : 0
% 69.10/9.25 # Total rewrite steps : 24581
% 69.10/9.25 # ...of those cached : 24514
% 69.10/9.25 # Propositional unsat checks : 0
% 69.10/9.25 # Propositional check models : 0
% 69.10/9.25 # Propositional check unsatisfiable : 0
% 69.10/9.25 # Propositional clauses : 0
% 69.10/9.25 # Propositional clauses after purity: 0
% 69.10/9.25 # Propositional unsat core size : 0
% 69.10/9.25 # Propositional preprocessing time : 0.000
% 69.10/9.25 # Propositional encoding time : 0.000
% 69.10/9.25 # Propositional solver time : 0.000
% 69.10/9.25 # Success case prop preproc time : 0.000
% 69.10/9.25 # Success case prop encoding time : 0.000
% 69.10/9.25 # Success case prop solver time : 0.000
% 69.10/9.25 # Current number of processed clauses : 956
% 69.10/9.25 # Positive orientable unit clauses : 38
% 69.10/9.25 # Positive unorientable unit clauses: 0
% 69.10/9.25 # Negative unit clauses : 20
% 69.10/9.25 # Non-unit-clauses : 898
% 69.10/9.25 # Current number of unprocessed clauses: 10229
% 69.10/9.25 # ...number of literals in the above : 79922
% 69.10/9.25 # Current number of archived formulas : 0
% 69.10/9.25 # Current number of archived clauses : 1935
% 69.10/9.25 # Clause-clause subsumption calls (NU) : 3202441
% 69.10/9.25 # Rec. Clause-clause subsumption calls : 189400
% 69.10/9.25 # Non-unit clause-clause subsumptions : 5275
% 69.10/9.25 # Unit Clause-clause subsumption calls : 21597
% 69.10/9.25 # Rewrite failures with RHS unbound : 0
% 69.10/9.25 # BW rewrite match attempts : 277
% 69.10/9.25 # BW rewrite match successes : 55
% 69.10/9.25 # Condensation attempts : 6962
% 69.10/9.25 # Condensation successes : 2
% 69.10/9.25 # Termbank termtop insertions : 5998745
% 69.10/9.25 # Search garbage collected termcells : 89079
% 69.10/9.25
% 69.10/9.25 # -------------------------------------------------
% 69.10/9.25 # User time : 8.525 s
% 69.10/9.25 # System time : 0.079 s
% 69.10/9.25 # Total time : 8.605 s
% 69.10/9.25 # Maximum resident set size: 30016 pages
% 69.10/9.25
% 69.10/9.25 # -------------------------------------------------
% 69.10/9.25 # User time : 8.539 s
% 69.10/9.25 # System time : 0.099 s
% 69.10/9.25 # Total time : 8.638 s
% 69.10/9.25 # Maximum resident set size: 7508 pages
% 69.10/9.25 % E---3.1 exiting
% 69.10/9.25 % E exiting
%------------------------------------------------------------------------------