TPTP Problem File: ITP086^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP086^2 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer KnowledgeKeysSecrets problem prob_194__3293238_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : KnowledgeKeysSecrets/prob_194__3293238_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.2.0, 0.00 v7.5.0
% Syntax : Number of formulae : 359 ( 119 unt; 56 typ; 0 def)
% Number of atoms : 986 ( 269 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3969 ( 107 ~; 28 |; 89 &;3297 @)
% ( 0 <=>; 448 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 128 ( 128 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 52 usr; 5 con; 0-3 aty)
% Number of variables : 1029 ( 76 ^; 904 !; 23 ?;1029 :)
% ( 26 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:19:14.132
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Secrecy__types_OExpression,type,
secrecy_Expression: $tType ).
thf(ty_t_Secrecy__types_OspecID,type,
secrecy_specID: $tType ).
thf(ty_t_Secrecy__types_OchanID,type,
secrecy_chanID: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (51)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Secrecy_Oeout,type,
eout: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OeoutM,type,
eoutM: secrecy_specID > ( set @ secrecy_chanID ) > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OexprChannel,type,
exprChannel: secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oine,type,
ine: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OineM,type,
ineM: secrecy_specID > ( set @ secrecy_chanID ) > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oine__exprChannelSet,type,
ine_exprChannelSet: secrecy_specID > ( set @ secrecy_chanID ) > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oine__exprChannelSingle,type,
ine_ex1303305700Single: secrecy_specID > secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oout__exprChannelSet,type,
out_exprChannelSet: secrecy_specID > ( set @ secrecy_chanID ) > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oout__exprChannelSingle,type,
out_ex985543062Single: secrecy_specID > secrecy_chanID > secrecy_Expression > $o ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_inout_OcorrectCompositionIn,type,
correctCompositionIn: secrecy_specID > $o ).
thf(sy_c_inout_OcorrectCompositionLoc,type,
correc965049635ionLoc: secrecy_specID > $o ).
thf(sy_c_inout_OcorrectCompositionOut,type,
correc990227761ionOut: secrecy_specID > $o ).
thf(sy_c_inout_OcorrectInOutLoc,type,
correctInOutLoc: secrecy_specID > $o ).
thf(sy_c_inout_OinStream,type,
inStream: secrecy_specID > ( set @ secrecy_chanID ) > $o ).
thf(sy_c_inout_Oins,type,
ins: secrecy_specID > ( set @ secrecy_chanID ) ).
thf(sy_c_inout_Oloc,type,
loc: secrecy_specID > ( set @ secrecy_chanID ) ).
thf(sy_c_inout_OlocStream,type,
locStream: secrecy_specID > ( set @ secrecy_chanID ) > $o ).
thf(sy_c_inout_Oout,type,
out: secrecy_specID > ( set @ secrecy_chanID ) ).
thf(sy_c_inout_OoutStream,type,
outStream: secrecy_specID > ( set @ secrecy_chanID ) > $o ).
thf(sy_c_inout_Osubcomponents,type,
subcomponents: secrecy_specID > ( set @ secrecy_specID ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: secrecy_specID ).
thf(sy_v_PQ,type,
pq: secrecy_specID ).
thf(sy_v_Q,type,
q: secrecy_specID ).
thf(sy_v_ch,type,
ch: secrecy_chanID ).
thf(sy_v_m,type,
m: secrecy_Expression ).
% Relevant facts (256)
thf(fact_0_assms_I2_J,axiom,
correctCompositionIn @ pq ).
% assms(2)
thf(fact_1_assms_I4_J,axiom,
exprChannel @ ch @ m ).
% assms(4)
thf(fact_2_assms_I3_J,axiom,
member @ secrecy_chanID @ ch @ ( ins @ p ) ).
% assms(3)
thf(fact_3_assms_I5_J,axiom,
! [X: secrecy_chanID] :
( ( member @ secrecy_chanID @ X @ ( ins @ pq ) )
=> ~ ( exprChannel @ X @ m ) ) ).
% assms(5)
thf(fact_4_assms_I1_J,axiom,
( ( subcomponents @ pq )
= ( insert @ secrecy_specID @ p @ ( insert @ secrecy_specID @ q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) ) ).
% assms(1)
thf(fact_5_locStream__def,axiom,
( locStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y: set @ secrecy_chanID,Z: set @ secrecy_chanID] : ( Y = Z )
@ ( loc @ X2 ) ) ) ) ).
% locStream_def
thf(fact_6_correctCompositionIn__L1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ~ ( member @ secrecy_chanID @ Ch @ ( loc @ PQ ) )
=> ( ( member @ secrecy_chanID @ Ch @ ( ins @ P ) )
=> ( member @ secrecy_chanID @ Ch @ ( ins @ PQ ) ) ) ) ) ) ).
% correctCompositionIn_L1
thf(fact_7_correctCompositionIn__L2,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,Ch: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member @ secrecy_chanID @ Ch @ ( ins @ PQ ) )
=> ( ( member @ secrecy_chanID @ Ch @ ( ins @ P ) )
| ( member @ secrecy_chanID @ Ch @ ( ins @ Q ) ) ) ) ) ) ).
% correctCompositionIn_L2
thf(fact_8_correctCompositionIn__prop1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,X3: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member @ secrecy_chanID @ X3 @ ( ins @ PQ ) )
=> ( ( member @ secrecy_chanID @ X3 @ ( ins @ P ) )
| ( member @ secrecy_chanID @ X3 @ ( ins @ Q ) ) ) ) ) ) ).
% correctCompositionIn_prop1
thf(fact_9_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_10_TBtheorem4b__P,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( ineM @ P @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( member @ secrecy_chanID @ Ch2 @ ( ins @ Q ) )
& ( exprChannel @ Ch2 @ E )
& ~ ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) )
& ( member @ secrecy_chanID @ Ch2 @ M ) )
=> ( ineM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheorem4b_P
thf(fact_11_TBtheorem4b__PQ,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( ineM @ P @ M @ E )
| ( ineM @ Q @ M @ E ) )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member @ secrecy_chanID @ Ch2 @ ( ins @ P ) )
| ( member @ secrecy_chanID @ Ch2 @ ( ins @ Q ) ) )
& ( member @ secrecy_chanID @ Ch2 @ M )
& ( exprChannel @ Ch2 @ E )
& ~ ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( ineM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheorem4b_PQ
thf(fact_12_TBlemma3b,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID,Ch: secrecy_chanID] :
( ~ ( ineM @ P @ M @ E )
=> ( ~ ( ineM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( member @ secrecy_chanID @ Ch @ M )
=> ( ( member @ secrecy_chanID @ Ch @ ( ins @ PQ ) )
=> ~ ( exprChannel @ Ch @ E ) ) ) ) ) ) ) ).
% TBlemma3b
thf(fact_13_inStream__def,axiom,
( inStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y: set @ secrecy_chanID,Z: set @ secrecy_chanID] : ( Y = Z )
@ ( ins @ X2 ) ) ) ) ).
% inStream_def
thf(fact_14_insertCI,axiom,
! [A: $tType,A2: A,B: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B ) ) ) ).
% insertCI
thf(fact_15_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_16_insert__absorb2,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) )
= ( insert @ A @ X3 @ A3 ) ) ).
% insert_absorb2
thf(fact_17_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_18_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_19_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_20_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X2: A] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_21_ineM__def,axiom,
( ineM
= ( ^ [SP: secrecy_specID,M2: set @ secrecy_chanID,E2: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member @ secrecy_chanID @ Ch3 @ ( ins @ SP ) )
& ( member @ secrecy_chanID @ Ch3 @ M2 )
& ( exprChannel @ Ch3 @ E2 ) ) ) ) ).
% ineM_def
thf(fact_22_ineM__L1,axiom,
! [Ch: secrecy_chanID,M: set @ secrecy_chanID,P: secrecy_specID,E: secrecy_Expression] :
( ( member @ secrecy_chanID @ Ch @ M )
=> ( ( member @ secrecy_chanID @ Ch @ ( ins @ P ) )
=> ( ( exprChannel @ Ch @ E )
=> ( ineM @ P @ M @ E ) ) ) ) ).
% ineM_L1
thf(fact_23_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_24_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_25_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_26_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_27_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B3: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B3 ) )
& ~ ( member @ A @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_28_insert__commute,axiom,
! [A: $tType,X3: A,Y3: A,A3: set @ A] :
( ( insert @ A @ X3 @ ( insert @ A @ Y3 @ A3 ) )
= ( insert @ A @ Y3 @ ( insert @ A @ X3 @ A3 ) ) ) ).
% insert_commute
thf(fact_29_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A,B: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B2 @ B )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A3 = B ) )
& ( ( A2 != B2 )
=> ? [C2: set @ A] :
( ( A3
= ( insert @ A @ B2 @ C2 ) )
& ~ ( member @ A @ B2 @ C2 )
& ( B
= ( insert @ A @ A2 @ C2 ) )
& ~ ( member @ A @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_30_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_31_insert__ident,axiom,
! [A: $tType,X3: A,A3: set @ A,B: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ~ ( member @ A @ X3 @ B )
=> ( ( ( insert @ A @ X3 @ A3 )
= ( insert @ A @ X3 @ B ) )
= ( A3 = B ) ) ) ) ).
% insert_ident
thf(fact_32_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ~ ! [B3: set @ A] :
( ( A3
= ( insert @ A @ X3 @ B3 ) )
=> ( member @ A @ X3 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_33_insertI2,axiom,
! [A: $tType,A2: A,B: set @ A,B2: A] :
( ( member @ A @ A2 @ B )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B ) ) ) ).
% insertI2
thf(fact_34_insertI1,axiom,
! [A: $tType,A2: A,B: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B ) ) ).
% insertI1
thf(fact_35_insertE,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_36_TBtheorem3b,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( ineM @ P @ M @ E )
=> ( ~ ( ineM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ~ ( ineM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheorem3b
thf(fact_37_TBtheorem1b,axiom,
! [PQ: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( ineM @ PQ @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ineM @ P @ M @ E )
| ( ineM @ Q @ M @ E ) ) ) ) ) ).
% TBtheorem1b
thf(fact_38_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_39_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_40_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C: A,D: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_41_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_42_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_43_TBtheorem4b__notP1,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ineM @ P @ M @ E )
=> ( ~ ( ineM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ine_ex1303305700Single @ P @ Ch2 @ E )
& ( member @ secrecy_chanID @ Ch2 @ M )
& ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( ineM @ PQ @ M @ E ) ) ) ) ) ) ).
% TBtheorem4b_notP1
thf(fact_44_TBtheorem4b__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set @ secrecy_chanID,E: secrecy_Expression,ChSetQ: set @ secrecy_chanID,M: set @ secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine_exprChannelSet @ P @ ChSetP @ E )
=> ( ( ine_exprChannelSet @ Q @ ChSetQ @ E )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSetP )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSetQ )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ~ ( ineM @ PQ @ M @ E ) ) ) ) ) ) ) ).
% TBtheorem4b_notPQ
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B4: $tType,A: $tType,F: A > B4,G: A > B4] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_TBtheorem4b__notP2,axiom,
! [Q: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set @ secrecy_chanID] :
( ~ ( ineM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine_exprChannelSet @ P @ ChSet @ E )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSet )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ~ ( ineM @ PQ @ M @ E ) ) ) ) ) ) ).
% TBtheorem4b_notP2
thf(fact_50_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_51_bot__apply,axiom,
! [C3: $tType,D2: $tType] :
( ( bot @ C3 )
=> ( ( bot_bot @ ( D2 > C3 ) )
= ( ^ [X2: D2] : ( bot_bot @ C3 ) ) ) ) ).
% bot_apply
thf(fact_52_TBtheorem4a__PQ,axiom,
! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( ine @ P @ E )
| ( ine @ Q @ E ) )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member @ secrecy_chanID @ Ch2 @ ( ins @ P ) )
| ( member @ secrecy_chanID @ Ch2 @ ( ins @ Q ) ) )
& ( exprChannel @ Ch2 @ E )
& ~ ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( ine @ PQ @ E ) ) ) ) ) ).
% TBtheorem4a_PQ
thf(fact_53_TBtheorem4a__P,axiom,
! [P: secrecy_specID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( ine @ P @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( member @ secrecy_chanID @ Ch2 @ ( ins @ P ) )
& ( exprChannel @ Ch2 @ E )
& ~ ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( ine @ PQ @ E ) ) ) ) ) ).
% TBtheorem4a_P
thf(fact_54_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_55_TBtheorem4a__empty,axiom,
! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( ine @ P @ E )
| ( ine @ Q @ E ) )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ( loc @ PQ )
= ( bot_bot @ ( set @ secrecy_chanID ) ) )
=> ( ine @ PQ @ E ) ) ) ) ) ).
% TBtheorem4a_empty
thf(fact_56_ine__exprChannelSingle__def,axiom,
( ine_ex1303305700Single
= ( ^ [SP: secrecy_specID,Ch3: secrecy_chanID,E2: secrecy_Expression] :
( ( member @ secrecy_chanID @ Ch3 @ ( ins @ SP ) )
& ( exprChannel @ Ch3 @ E2 )
& ! [X2: secrecy_chanID,T: nat] :
( ( ( member @ secrecy_chanID @ X2 @ ( ins @ SP ) )
& ( X2 != Ch3 ) )
=> ~ ( exprChannel @ X2 @ E2 ) ) ) ) ) ).
% ine_exprChannelSingle_def
thf(fact_57_ine__empty__exprChannelSet,axiom,
! [P: secrecy_specID,ChSet: set @ secrecy_chanID,E: secrecy_Expression] :
( ( ine_exprChannelSet @ P @ ChSet @ E )
=> ( ( ChSet
= ( bot_bot @ ( set @ secrecy_chanID ) ) )
=> ~ ( ine @ P @ E ) ) ) ).
% ine_empty_exprChannelSet
thf(fact_58_ine__exprChannelSet__Single,axiom,
! [P: secrecy_specID,Ch: secrecy_chanID,E: secrecy_Expression] :
( ( ine_exprChannelSet @ P @ ( insert @ secrecy_chanID @ Ch @ ( bot_bot @ ( set @ secrecy_chanID ) ) ) @ E )
=> ( ine_ex1303305700Single @ P @ Ch @ E ) ) ).
% ine_exprChannelSet_Single
thf(fact_59_ine__exprChannelSingle__Set,axiom,
! [P: secrecy_specID,Ch: secrecy_chanID,E: secrecy_Expression] :
( ( ine_ex1303305700Single @ P @ Ch @ E )
=> ( ine_exprChannelSet @ P @ ( insert @ secrecy_chanID @ Ch @ ( bot_bot @ ( set @ secrecy_chanID ) ) ) @ E ) ) ).
% ine_exprChannelSingle_Set
thf(fact_60_ine__nonempty__exprChannelSet,axiom,
! [P: secrecy_specID,ChSet: set @ secrecy_chanID,E: secrecy_Expression] :
( ( ine_exprChannelSet @ P @ ChSet @ E )
=> ( ( ChSet
!= ( bot_bot @ ( set @ secrecy_chanID ) ) )
=> ( ine @ P @ E ) ) ) ).
% ine_nonempty_exprChannelSet
thf(fact_61_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_62_not__ine__ineM,axiom,
! [P: secrecy_specID,E: secrecy_Expression,M: set @ secrecy_chanID] :
( ~ ( ine @ P @ E )
=> ~ ( ineM @ P @ M @ E ) ) ).
% not_ine_ineM
thf(fact_63_ineM__ine,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression] :
( ( ineM @ P @ M @ E )
=> ( ine @ P @ E ) ) ).
% ineM_ine
thf(fact_64_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
( A4
= ( insert @ A @ ( the_elem @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_65_ine__def,axiom,
( ine
= ( ^ [SP: secrecy_specID,E2: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member @ secrecy_chanID @ Ch3 @ ( ins @ SP ) )
& ( exprChannel @ Ch3 @ E2 ) ) ) ) ).
% ine_def
thf(fact_66_ine__ins__neg1,axiom,
! [P: secrecy_specID,M3: secrecy_Expression,X3: secrecy_chanID] :
( ~ ( ine @ P @ M3 )
=> ( ( exprChannel @ X3 @ M3 )
=> ~ ( member @ secrecy_chanID @ X3 @ ( ins @ P ) ) ) ) ).
% ine_ins_neg1
thf(fact_67_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y2: A] :
( ( member @ A @ X4 @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( X4 = Y2 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_68_bot__fun__def,axiom,
! [B4: $tType,A: $tType] :
( ( bot @ B4 )
=> ( ( bot_bot @ ( A > B4 ) )
= ( ^ [X2: A] : ( bot_bot @ B4 ) ) ) ) ).
% bot_fun_def
thf(fact_69_TBtheorem4a__notP2,axiom,
! [Q: secrecy_specID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set @ secrecy_chanID] :
( ~ ( ine @ Q @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine_exprChannelSet @ P @ ChSet @ E )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSet )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ~ ( ine @ PQ @ E ) ) ) ) ) ) ).
% TBtheorem4a_notP2
thf(fact_70_TBtheorem4a__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set @ secrecy_chanID,E: secrecy_Expression,ChSetQ: set @ secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine_exprChannelSet @ P @ ChSetP @ E )
=> ( ( ine_exprChannelSet @ Q @ ChSetQ @ E )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSetP )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSetQ )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ~ ( ine @ PQ @ E ) ) ) ) ) ) ) ).
% TBtheorem4a_notPQ
thf(fact_71_TBtheorem4a__notP1,axiom,
! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ine @ P @ E )
=> ( ~ ( ine @ Q @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ine_ex1303305700Single @ P @ Ch2 @ E )
& ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( ine @ PQ @ E ) ) ) ) ) ) ).
% TBtheorem4a_notP1
thf(fact_72_ine__exprChannelSet__def,axiom,
( ine_exprChannelSet
= ( ^ [SP: secrecy_specID,ChSet2: set @ secrecy_chanID,E2: secrecy_Expression] :
( ! [X2: secrecy_chanID] :
( ( member @ secrecy_chanID @ X2 @ ChSet2 )
=> ( ( member @ secrecy_chanID @ X2 @ ( ins @ SP ) )
& ( exprChannel @ X2 @ E2 ) ) )
& ! [X2: secrecy_chanID] :
( ( ~ ( member @ secrecy_chanID @ X2 @ ChSet2 )
& ( member @ secrecy_chanID @ X2 @ ( ins @ SP ) ) )
=> ~ ( exprChannel @ X2 @ E2 ) ) ) ) ) ).
% ine_exprChannelSet_def
thf(fact_73_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X4: A] :
( A3
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_74_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A4: set @ A] :
? [X2: A] :
( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_75_TBtheorem1a,axiom,
! [PQ: secrecy_specID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( ine @ PQ @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ( ( ine @ P @ E )
| ( ine @ Q @ E ) ) ) ) ) ).
% TBtheorem1a
thf(fact_76_TBtheorem3a,axiom,
! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ~ ( ine @ P @ E )
=> ( ~ ( ine @ Q @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correctCompositionIn @ PQ )
=> ~ ( ine @ PQ @ E ) ) ) ) ) ).
% TBtheorem3a
thf(fact_77_subcomponents__loc,axiom,
! [X3: secrecy_specID] :
( ( correc965049635ionLoc @ X3 )
=> ( ( ( subcomponents @ X3 )
= ( bot_bot @ ( set @ secrecy_specID ) ) )
=> ( ( loc @ X3 )
= ( bot_bot @ ( set @ secrecy_chanID ) ) ) ) ) ).
% subcomponents_loc
thf(fact_78_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_79_TBtheorem5a__empty,axiom,
! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( eout @ P @ E )
| ( eout @ Q @ E ) )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( ( loc @ PQ )
= ( bot_bot @ ( set @ secrecy_chanID ) ) )
=> ( eout @ PQ @ E ) ) ) ) ) ).
% TBtheorem5a_empty
thf(fact_80_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_81_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_82_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A3: A] : ( pairwise @ A @ P @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_83_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_84_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_85_subsetI,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ).
% subsetI
thf(fact_86_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ A3 )
=> ( A3 = B ) ) ) ).
% subset_antisym
thf(fact_87_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_88_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_89_insert__subset,axiom,
! [A: $tType,X3: A,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B )
= ( ( member @ A @ X3 @ B )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ) ).
% insert_subset
thf(fact_90_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_91_le__funD,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ! [F: A > B4,G: A > B4,X3: A] :
( ( ord_less_eq @ ( A > B4 ) @ F @ G )
=> ( ord_less_eq @ B4 @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_92_le__funE,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ! [F: A > B4,G: A > B4,X3: A] :
( ( ord_less_eq @ ( A > B4 ) @ F @ G )
=> ( ord_less_eq @ B4 @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_93_le__funI,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ! [F: A > B4,G: A > B4] :
( ! [X4: A] : ( ord_less_eq @ B4 @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B4 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_94_le__fun__def,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ( ( ord_less_eq @ ( A > B4 ) )
= ( ^ [F2: A > B4,G2: A > B4] :
! [X2: A] : ( ord_less_eq @ B4 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_95_order__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 )
& ( order @ A ) )
=> ! [A2: A,F: B4 > A,B2: B4,C: B4] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B4 @ B2 @ C )
=> ( ! [X4: B4,Y2: B4] :
( ( ord_less_eq @ B4 @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_96_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_97_ord__eq__le__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 )
& ( ord @ A ) )
=> ! [A2: A,F: B4 > A,B2: B4,C: B4] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B4 @ B2 @ C )
=> ( ! [X4: B4,Y2: B4] :
( ( ord_less_eq @ B4 @ X4 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_98_ord__le__eq__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B4,C: B4] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ B4 @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_100_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% antisym
thf(fact_101_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linear
thf(fact_102_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% eq_refl
thf(fact_103_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% le_cases
thf(fact_104_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_105_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_106_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_107_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_108_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_109_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_110_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_111_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_112_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_113_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_114_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_115_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_116_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_117_in__mono,axiom,
! [A: $tType,A3: set @ A,B: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B ) ) ) ).
% in_mono
thf(fact_118_subsetD,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B ) ) ) ).
% subsetD
thf(fact_119_equalityE,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( A3 = B )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ~ ( ord_less_eq @ ( set @ A ) @ B @ A3 ) ) ) ).
% equalityE
thf(fact_120_pairwiseD,axiom,
! [A: $tType,R: A > A > $o,S: set @ A,X3: A,Y3: A] :
( ( pairwise @ A @ R @ S )
=> ( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_121_pairwiseI,axiom,
! [A: $tType,S: set @ A,R: A > A > $o] :
( ! [X4: A,Y2: A] :
( ( member @ A @ X4 @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( ( X4 != Y2 )
=> ( R @ X4 @ Y2 ) ) ) )
=> ( pairwise @ A @ R @ S ) ) ).
% pairwiseI
thf(fact_122_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_123_equalityD1,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( A3 = B )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ).
% equalityD1
thf(fact_124_equalityD2,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( A3 = B )
=> ( ord_less_eq @ ( set @ A ) @ B @ A3 ) ) ).
% equalityD2
thf(fact_125_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B7: set @ A] :
! [T: A] :
( ( member @ A @ T @ A4 )
=> ( member @ A @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_126_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_127_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_128_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S2: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ( X2 != Y4 )
=> ( R2 @ X2 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_129_subset__trans,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% subset_trans
thf(fact_130_pairwise__mono,axiom,
! [A: $tType,P: A > A > $o,A3: set @ A,Q: A > A > $o,B: set @ A] :
( ( pairwise @ A @ P @ A3 )
=> ( ! [X4: A,Y2: A] :
( ( P @ X4 @ Y2 )
=> ( Q @ X4 @ Y2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ A3 )
=> ( pairwise @ A @ Q @ B ) ) ) ) ).
% pairwise_mono
thf(fact_131_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
= ( ^ [A4: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_132_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T2: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( pairwise @ A @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_133_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_134_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_135_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_136_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_137_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_138_subset__insertI,axiom,
! [A: $tType,B: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B @ ( insert @ A @ A2 @ B ) ) ).
% subset_insertI
thf(fact_139_subset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A,B: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B ) ) ) ).
% subset_insert
thf(fact_140_insert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_141_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_142_pairwise__insert,axiom,
! [A: $tType,R3: A > A > $o,X3: A,S3: set @ A] :
( ( pairwise @ A @ R3 @ ( insert @ A @ X3 @ S3 ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S3 )
& ( Y4 != X3 ) )
=> ( ( R3 @ X3 @ Y4 )
& ( R3 @ Y4 @ X3 ) ) )
& ( pairwise @ A @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_143_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_144_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_145_TBtheorem2a,axiom,
! [PQ: secrecy_specID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( eout @ PQ @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( eout @ P @ E )
| ( eout @ Q @ E ) ) ) ) ) ).
% TBtheorem2a
thf(fact_146_TBtheorem5a__notP1,axiom,
! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( eout @ P @ E )
=> ( ~ ( eout @ Q @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( out_ex985543062Single @ P @ Ch2 @ E )
& ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( eout @ PQ @ E ) ) ) ) ) ) ).
% TBtheorem5a_notP1
thf(fact_147_TBtheorem5a__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set @ secrecy_chanID,E: secrecy_Expression,ChSetQ: set @ secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSetP @ E )
=> ( ( out_exprChannelSet @ Q @ ChSetQ @ E )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSetP )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSetQ )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ~ ( eout @ PQ @ E ) ) ) ) ) ) ) ).
% TBtheorem5a_notPQ
thf(fact_148_TBtheorem5a__notP2,axiom,
! [Q: secrecy_specID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set @ secrecy_chanID] :
( ~ ( eout @ Q @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSet @ E )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSet )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ~ ( eout @ PQ @ E ) ) ) ) ) ) ).
% TBtheorem5a_notP2
thf(fact_149_TBtheorem5a__PQ,axiom,
! [P: secrecy_specID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( eout @ P @ E )
| ( eout @ Q @ E ) )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member @ secrecy_chanID @ Ch2 @ ( out @ P ) )
| ( member @ secrecy_chanID @ Ch2 @ ( out @ Q ) ) )
& ( exprChannel @ Ch2 @ E )
& ~ ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( eout @ PQ @ E ) ) ) ) ) ).
% TBtheorem5a_PQ
thf(fact_150_TBtheorem45a__P,axiom,
! [P: secrecy_specID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( eout @ P @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( member @ secrecy_chanID @ Ch2 @ ( out @ P ) )
& ( exprChannel @ Ch2 @ E )
& ~ ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( eout @ PQ @ E ) ) ) ) ) ).
% TBtheorem45a_P
thf(fact_151_TBtheorem2b,axiom,
! [PQ: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,P: secrecy_specID,Q: secrecy_specID] :
( ( eoutM @ PQ @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( eoutM @ P @ M @ E )
| ( eoutM @ Q @ M @ E ) ) ) ) ) ).
% TBtheorem2b
thf(fact_152_eoutM__def,axiom,
( eoutM
= ( ^ [SP: secrecy_specID,M2: set @ secrecy_chanID,E2: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member @ secrecy_chanID @ Ch3 @ ( out @ SP ) )
& ( member @ secrecy_chanID @ Ch3 @ M2 )
& ( exprChannel @ Ch3 @ E2 ) ) ) ) ).
% eoutM_def
thf(fact_153_out__exprChannelSet__def,axiom,
( out_exprChannelSet
= ( ^ [SP: secrecy_specID,ChSet2: set @ secrecy_chanID,E2: secrecy_Expression] :
( ! [X2: secrecy_chanID] :
( ( member @ secrecy_chanID @ X2 @ ChSet2 )
=> ( ( member @ secrecy_chanID @ X2 @ ( out @ SP ) )
& ( exprChannel @ X2 @ E2 ) ) )
& ! [X2: secrecy_chanID] :
( ( ~ ( member @ secrecy_chanID @ X2 @ ChSet2 )
& ( member @ secrecy_chanID @ X2 @ ( out @ SP ) ) )
=> ~ ( exprChannel @ X2 @ E2 ) ) ) ) ) ).
% out_exprChannelSet_def
thf(fact_154_out__exprChannelSingle__def,axiom,
( out_ex985543062Single
= ( ^ [SP: secrecy_specID,Ch3: secrecy_chanID,E2: secrecy_Expression] :
( ( member @ secrecy_chanID @ Ch3 @ ( out @ SP ) )
& ( exprChannel @ Ch3 @ E2 )
& ! [X2: secrecy_chanID,T: nat] :
( ( ( member @ secrecy_chanID @ X2 @ ( out @ SP ) )
& ( X2 != Ch3 ) )
=> ~ ( exprChannel @ X2 @ E2 ) ) ) ) ) ).
% out_exprChannelSingle_def
thf(fact_155_not__eout__eoutM,axiom,
! [P: secrecy_specID,E: secrecy_Expression,M: set @ secrecy_chanID] :
( ~ ( eout @ P @ E )
=> ~ ( eoutM @ P @ M @ E ) ) ).
% not_eout_eoutM
thf(fact_156_eoutM__eout,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression] :
( ( eoutM @ P @ M @ E )
=> ( eout @ P @ E ) ) ).
% eoutM_eout
thf(fact_157_out__exprChannelSet__Single,axiom,
! [P: secrecy_specID,Ch: secrecy_chanID,E: secrecy_Expression] :
( ( out_exprChannelSet @ P @ ( insert @ secrecy_chanID @ Ch @ ( bot_bot @ ( set @ secrecy_chanID ) ) ) @ E )
=> ( out_ex985543062Single @ P @ Ch @ E ) ) ).
% out_exprChannelSet_Single
thf(fact_158_out__exprChannelSingle__Set,axiom,
! [P: secrecy_specID,Ch: secrecy_chanID,E: secrecy_Expression] :
( ( out_ex985543062Single @ P @ Ch @ E )
=> ( out_exprChannelSet @ P @ ( insert @ secrecy_chanID @ Ch @ ( bot_bot @ ( set @ secrecy_chanID ) ) ) @ E ) ) ).
% out_exprChannelSingle_Set
thf(fact_159_eout__def,axiom,
( eout
= ( ^ [SP: secrecy_specID,E2: secrecy_Expression] :
? [Ch3: secrecy_chanID] :
( ( member @ secrecy_chanID @ Ch3 @ ( out @ SP ) )
& ( exprChannel @ Ch3 @ E2 ) ) ) ) ).
% eout_def
thf(fact_160_TBtheorem5b__notP2,axiom,
! [Q: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,P: secrecy_specID,ChSet: set @ secrecy_chanID] :
( ~ ( eoutM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSet @ E )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSet )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ~ ( eoutM @ PQ @ M @ E ) ) ) ) ) ) ).
% TBtheorem5b_notP2
thf(fact_161_TBtheore54b__P,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,PQ: secrecy_specID,Q: secrecy_specID] :
( ( eoutM @ P @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( member @ secrecy_chanID @ Ch2 @ ( out @ Q ) )
& ( exprChannel @ Ch2 @ E )
& ~ ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) )
& ( member @ secrecy_chanID @ Ch2 @ M ) )
=> ( eoutM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheore54b_P
thf(fact_162_TBtheorem5b__PQ,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( ( eoutM @ P @ M @ E )
| ( eoutM @ Q @ M @ E ) )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( ( member @ secrecy_chanID @ Ch2 @ ( out @ P ) )
| ( member @ secrecy_chanID @ Ch2 @ ( out @ Q ) ) )
& ( member @ secrecy_chanID @ Ch2 @ M )
& ( exprChannel @ Ch2 @ E )
& ~ ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ( eoutM @ PQ @ M @ E ) ) ) ) ) ).
% TBtheorem5b_PQ
thf(fact_163_TBtheorem5b__notP1,axiom,
! [P: secrecy_specID,M: set @ secrecy_chanID,E: secrecy_Expression,Q: secrecy_specID,PQ: secrecy_specID] :
( ( eoutM @ P @ M @ E )
=> ( ~ ( eoutM @ Q @ M @ E )
=> ( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ? [Ch2: secrecy_chanID] :
( ( out_ex985543062Single @ P @ Ch2 @ E )
& ( member @ secrecy_chanID @ Ch2 @ M )
& ( member @ secrecy_chanID @ Ch2 @ ( loc @ PQ ) ) )
=> ~ ( eoutM @ PQ @ M @ E ) ) ) ) ) ) ).
% TBtheorem5b_notP1
thf(fact_164_correctCompositionOut__prop1,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,X3: secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( member @ secrecy_chanID @ X3 @ ( out @ PQ ) )
=> ( ( member @ secrecy_chanID @ X3 @ ( out @ P ) )
| ( member @ secrecy_chanID @ X3 @ ( out @ Q ) ) ) ) ) ) ).
% correctCompositionOut_prop1
thf(fact_165_outStream__def,axiom,
( outStream
= ( ^ [X2: secrecy_specID] :
( ^ [Y: set @ secrecy_chanID,Z: set @ secrecy_chanID] : ( Y = Z )
@ ( out @ X2 ) ) ) ) ).
% outStream_def
thf(fact_166_TBtheorem5b__notPQ,axiom,
! [PQ: secrecy_specID,P: secrecy_specID,Q: secrecy_specID,ChSetP: set @ secrecy_chanID,E: secrecy_Expression,ChSetQ: set @ secrecy_chanID,M: set @ secrecy_chanID] :
( ( ( subcomponents @ PQ )
= ( insert @ secrecy_specID @ P @ ( insert @ secrecy_specID @ Q @ ( bot_bot @ ( set @ secrecy_specID ) ) ) ) )
=> ( ( correc990227761ionOut @ PQ )
=> ( ( out_exprChannelSet @ P @ ChSetP @ E )
=> ( ( out_exprChannelSet @ Q @ ChSetQ @ E )
=> ( ( M
= ( sup_sup @ ( set @ secrecy_chanID ) @ ChSetP @ ChSetQ ) )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSetP )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ( ! [X4: secrecy_chanID] :
( ( member @ secrecy_chanID @ X4 @ ChSetQ )
=> ( member @ secrecy_chanID @ X4 @ ( loc @ PQ ) ) )
=> ~ ( eoutM @ PQ @ M @ E ) ) ) ) ) ) ) ) ).
% TBtheorem5b_notPQ
thf(fact_167_insert__subsetI,axiom,
! [A: $tType,X3: A,A3: set @ A,X5: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_168_UnCI,axiom,
! [A: $tType,C: A,B: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C @ B )
=> ( member @ A @ C @ A3 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B ) ) ) ).
% UnCI
thf(fact_169_Un__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B ) )
= ( ( member @ A @ C @ A3 )
| ( member @ A @ C @ B ) ) ) ).
% Un_iff
thf(fact_170_Un__empty,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_171_Un__insert__left,axiom,
! [A: $tType,A2: A,B: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B ) @ C4 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B @ C4 ) ) ) ).
% Un_insert_left
thf(fact_172_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A2: A,B: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A3 @ B ) ) ) ).
% Un_insert_right
thf(fact_173_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_174_Un__empty__left,axiom,
! [A: $tType,B: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B )
= B ) ).
% Un_empty_left
thf(fact_175_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_176_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A4: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_177_subset__UnE,axiom,
! [A: $tType,C4: set @ A,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ A3 @ B ) )
=> ~ ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ A3 )
=> ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ B )
=> ( C4
!= ( sup_sup @ ( set @ A ) @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_178_Un__absorb2,axiom,
! [A: $tType,B: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B )
= A3 ) ) ).
% Un_absorb2
thf(fact_179_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_180_Un__upper2,axiom,
! [A: $tType,B: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B @ ( sup_sup @ ( set @ A ) @ A3 @ B ) ) ).
% Un_upper2
thf(fact_181_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B ) ) ).
% Un_upper1
thf(fact_182_Un__least,axiom,
! [A: $tType,A3: set @ A,C4: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B ) @ C4 ) ) ) ).
% Un_least
thf(fact_183_Un__mono,axiom,
! [A: $tType,A3: set @ A,C4: set @ A,B: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B ) @ ( sup_sup @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_184_ineM__Un1,axiom,
! [P: secrecy_specID,A3: set @ secrecy_chanID,E: secrecy_Expression,B: set @ secrecy_chanID] :
( ( ineM @ P @ A3 @ E )
=> ( ineM @ P @ ( sup_sup @ ( set @ secrecy_chanID ) @ A3 @ B ) @ E ) ) ).
% ineM_Un1
thf(fact_185_UnE,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B ) )
=> ( ~ ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B ) ) ) ).
% UnE
thf(fact_186_UnI1,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B ) ) ) ).
% UnI1
thf(fact_187_UnI2,axiom,
! [A: $tType,C: A,B: set @ A,A3: set @ A] :
( ( member @ A @ C @ B )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B ) ) ) ).
% UnI2
thf(fact_188_bex__Un,axiom,
! [A: $tType,A3: set @ A,B: set @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_189_ball__Un,axiom,
! [A: $tType,A3: set @ A,B: set @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_190_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B @ C4 ) ) ) ).
% Un_assoc
thf(fact_191_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_192_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A4: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A4 ) ) ) ).
% Un_commute
thf(fact_193_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B ) ) ).
% Un_left_absorb
thf(fact_194_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B @ ( sup_sup @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_195_singleton__Un__iff,axiom,
! [A: $tType,X3: A,A3: set @ A,B: set @ A] :
( ( ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_196_Un__singleton__iff,axiom,
! [A: $tType,A3: set @ A,B: set @ A,X3: A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B )
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_197_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_198_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_199_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_200_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A2 @ B2 ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_201_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_202_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
& ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_203_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_204_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% sup_bot_left
thf(fact_205_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% sup_bot_right
thf(fact_206_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A,Y3: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X3 @ Y3 ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y3
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_207_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A,Y3: A] :
( ( ( sup_sup @ A @ X3 @ Y3 )
= ( bot_bot @ A ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_208_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( sup_sup @ A @ A2 @ B2 )
= ( bot_bot @ A ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_209_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y3: A,X3: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% inf_sup_ord(4)
thf(fact_210_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% inf_sup_ord(3)
thf(fact_211_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq @ A @ A2 @ X3 )
=> ~ ( ord_less_eq @ A @ B2 @ X3 ) ) ) ) ).
% le_supE
thf(fact_212_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
=> ( ( ord_less_eq @ A @ B2 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X3 ) ) ) ) ).
% le_supI
thf(fact_213_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% sup_ge1
thf(fact_214_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X3: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X3 @ Y3 ) ) ) ).
% sup_ge2
thf(fact_215_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_216_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_217_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C: A,A2: A,D: A,B2: A] :
( ( ord_less_eq @ A @ C @ A2 )
=> ( ( ord_less_eq @ A @ D @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C @ D ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_218_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C: A,B2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ( ord_less_eq @ A @ B2 @ D )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C @ D ) ) ) ) ) ).
% sup_mono
thf(fact_219_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X3: A,Z2: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y3 @ Z2 ) @ X3 ) ) ) ) ).
% sup_least
thf(fact_220_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y4: A] :
( ( sup_sup @ A @ X2 @ Y4 )
= Y4 ) ) ) ) ).
% le_iff_sup
thf(fact_221_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_222_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_223_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X3: A,Y3: A] :
( ! [X4: A,Y2: A] : ( ord_less_eq @ A @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ Y2 @ X4 )
=> ( ( ord_less_eq @ A @ Z3 @ X4 )
=> ( ord_less_eq @ A @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup @ A @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ) ).
% sup_unique
thf(fact_224_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_225_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_226_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( sup_sup @ A @ X3 @ Y3 )
= X3 ) ) ) ).
% sup_absorb1
thf(fact_227_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( sup_sup @ A @ X3 @ Y3 )
= Y3 ) ) ) ).
% sup_absorb2
thf(fact_228_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_229_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_230_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_231_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_232_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_233_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_234_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( sup_sup @ A @ A5 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_235_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C @ A2 )
=> ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_236_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_237_correctInOutLoc__def,axiom,
( correctInOutLoc
= ( ^ [X2: secrecy_specID] :
( ( ( inf_inf @ ( set @ secrecy_chanID ) @ ( ins @ X2 ) @ ( out @ X2 ) )
= ( bot_bot @ ( set @ secrecy_chanID ) ) )
& ( ( inf_inf @ ( set @ secrecy_chanID ) @ ( ins @ X2 ) @ ( loc @ X2 ) )
= ( bot_bot @ ( set @ secrecy_chanID ) ) )
& ( ( inf_inf @ ( set @ secrecy_chanID ) @ ( loc @ X2 ) @ ( out @ X2 ) )
= ( bot_bot @ ( set @ secrecy_chanID ) ) ) ) ) ) ).
% correctInOutLoc_def
thf(fact_238_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_239_IntI,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( member @ A @ C @ B )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B ) ) ) ) ).
% IntI
thf(fact_240_Int__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B ) )
= ( ( member @ A @ C @ A3 )
& ( member @ A @ C @ B ) ) ) ).
% Int_iff
thf(fact_241_ComplI,axiom,
! [A: $tType,C: A,A3: set @ A] :
( ~ ( member @ A @ C @ A3 )
=> ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_242_Compl__iff,axiom,
! [A: $tType,C: A,A3: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C @ A3 ) ) ) ).
% Compl_iff
thf(fact_243_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B ) )
= ( A3 = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_244_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% compl_le_compl_iff
thf(fact_245_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) )
= ( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_246_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% inf.bounded_iff
thf(fact_247_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_248_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_249_Int__insert__left__if0,axiom,
! [A: $tType,A2: A,C4: set @ A,B: set @ A] :
( ~ ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_250_Int__insert__left__if1,axiom,
! [A: $tType,A2: A,C4: set @ A,B: set @ A] :
( ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B ) @ C4 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_251_insert__inter__insert,axiom,
! [A: $tType,A2: A,A3: set @ A,B: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A3 ) @ ( insert @ A @ A2 @ B ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B ) ) ) ).
% insert_inter_insert
thf(fact_252_Int__insert__right__if0,axiom,
! [A: $tType,A2: A,A3: set @ A,B: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_253_Int__insert__right__if1,axiom,
! [A: $tType,A2: A,A3: set @ A,B: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A3 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_254_Int__subset__iff,axiom,
! [A: $tType,C4: set @ A,A3: set @ A,B: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A3 @ B ) )
= ( ( ord_less_eq @ ( set @ A ) @ C4 @ A3 )
& ( ord_less_eq @ ( set @ A ) @ C4 @ B ) ) ) ).
% Int_subset_iff
thf(fact_255_Int__Un__eq_I4_J,axiom,
! [A: $tType,T2: set @ A,S: set @ A] :
( ( sup_sup @ ( set @ A ) @ T2 @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
% Type constructors (46)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounde1808546759up_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 )
=> ( semilattice_sup @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_inf @ A9 )
=> ( semilattice_inf @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A8: $tType,A9: $tType] :
( ( boolean_algebra @ A9 )
=> ( boolean_algebra @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 )
=> ( order_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 )
=> ( lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_4,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_5,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_6,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_7,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_8,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_9,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_10,axiom,
bot @ nat ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_11,axiom,
! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_12,axiom,
! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_13,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_14,axiom,
! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_15,axiom,
! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_16,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_17,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_18,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_19,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_20,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_21,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_22,axiom,
bounde1808546759up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_23,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_24,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_25,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_26,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_27,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_28,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_29,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_30,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_31,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_32,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_33,axiom,
bot @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
member @ secrecy_chanID @ ch @ ( loc @ pq ) ).
%------------------------------------------------------------------------------