TPTP Problem File: PRO021^1.p
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%------------------------------------------------------------------------------
% File : PRO021^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Processes
% Problem : Process algebra 159
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : process__159.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 322 ( 137 unt; 60 typ; 0 def)
% Number of atoms : 846 ( 308 equ; 0 cnn)
% Maximal formula atoms : 31 ( 3 avg)
% Number of connectives : 6517 ( 106 ~; 3 |; 33 &;6055 @)
% ( 0 <=>; 320 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 12 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 1627 (1627 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 3 con; 0-14 aty)
% Number of variables : 1933 ( 153 ^;1671 !; 11 ?;1933 :)
% ( 98 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:39:50.418
%------------------------------------------------------------------------------
%----Could-be-implicit typings (8)
thf(ty_t_Process__Mirabelle__xovoopgpgi_Oprocess__term,type,
proces554692349s_term: $tType > $tType > $tType ).
thf(ty_t_Process__Mirabelle__xovoopgpgi_Oprocess,type,
proces634752977rocess: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (52)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple1396247847notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oguarded,type,
proces1121166967uarded:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > ( proces554692349s_term @ B @ C ) ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_OisACT,type,
proces687458811_isACT:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( proces554692349s_term @ B @ C ) ) > ( proces554692349s_term @ D @ A ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Ocase__process,type,
proces1406508781rocess:
!>[A: $tType,B: $tType] : ( ( A > ( proces634752977rocess @ A ) > B ) > ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B ) > ( proces634752977rocess @ A ) > B ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Ocorec__process,type,
proces894737309rocess:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Opred__process,type,
proces1205983068rocess:
!>[A: $tType] : ( ( A > $o ) > ( proces634752977rocess @ A ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Orel__process,type,
proces749077512rocess:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( proces634752977rocess @ A ) > ( proces634752977rocess @ B ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Oset__process,type,
proces1148692671rocess:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( set @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_OACT,type,
proces1454156180ss_ACT:
!>[A: $tType,Pvar: $tType] : ( A > ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_OCH,type,
proces89589571ess_CH:
!>[A: $tType,Pvar: $tType] : ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_OPROC,type,
proces1062592052s_PROC:
!>[A: $tType,Pvar: $tType] : ( ( proces634752977rocess @ A ) > ( proces554692349s_term @ A @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_OVAR,type,
proces1627516585ss_VAR:
!>[Pvar: $tType,A: $tType] : ( Pvar > ( proces554692349s_term @ A @ Pvar ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Ocase__process__term,type,
proces460752237s_term:
!>[Pvar: $tType,B: $tType,A: $tType] : ( ( Pvar > B ) > ( ( proces634752977rocess @ A ) > B ) > ( A > ( proces554692349s_term @ A @ Pvar ) > B ) > ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B ) > ( proces554692349s_term @ A @ Pvar ) > B ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Omap__process__term,type,
proces224683037s_term:
!>[A: $tType,Aa: $tType,Pvar: $tType,Pvara: $tType] : ( ( A > Aa ) > ( Pvar > Pvara ) > ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ Aa @ Pvara ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Opred__process__term,type,
proces2118920028s_term:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( proces554692349s_term @ A @ B ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Orec__process__term,type,
proces2117273769s_term:
!>[Pvar: $tType,D: $tType,A: $tType] : ( ( Pvar > D ) > ( ( proces634752977rocess @ A ) > D ) > ( A > ( proces554692349s_term @ A @ Pvar ) > D > D ) > ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D ) > ( proces554692349s_term @ A @ Pvar ) > D ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Orel__process__term,type,
proces2029722208s_term:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( proces554692349s_term @ A @ B ) > ( proces554692349s_term @ C @ D ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Oset1__process__term,type,
proces1264666542s_term:
!>[A: $tType,Pvar: $tType] : ( ( proces554692349s_term @ A @ Pvar ) > ( set @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess__term_Oset2__process__term,type,
proces1493547885s_term:
!>[A: $tType,Pvar: $tType] : ( ( proces554692349s_term @ A @ Pvar ) > ( set @ Pvar ) ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quotient_OBex1__rel,type,
bex1_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $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_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Orev__implies,type,
rev_implies: $o > $o > $o ).
thf(sy_c_Transfer_Otransfer__forall,type,
transfer_forall:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_X,type,
x: a ).
thf(sy_v_sys,type,
sys: a > ( proces554692349s_term @ b @ a ) ).
%----Relevant facts (255)
thf(fact_0_g,axiom,
proces1121166967uarded @ a @ b @ a @ sys ).
% g
thf(fact_1_process__term_Oinject_I1_J,axiom,
! [A: $tType,Pvar: $tType,X1: Pvar,Y1: Pvar] :
( ( ( proces1627516585ss_VAR @ Pvar @ A @ X1 )
= ( proces1627516585ss_VAR @ Pvar @ A @ Y1 ) )
= ( X1 = Y1 ) ) ).
% process_term.inject(1)
thf(fact_2_guarded__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( proces1121166967uarded @ A @ B @ C )
= ( ^ [Sys: A > ( proces554692349s_term @ B @ C )] :
! [X: A,Y: C] :
( ( Sys @ X )
!= ( proces1627516585ss_VAR @ C @ B @ Y ) ) ) ) ).
% guarded_def
thf(fact_3_process__term_Opred__inject_I1_J,axiom,
! [A: $tType,Pvar: $tType,P1: A > $o,P2: Pvar > $o,A2: Pvar] :
( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ ( proces1627516585ss_VAR @ Pvar @ A @ A2 ) )
= ( P2 @ A2 ) ) ).
% process_term.pred_inject(1)
thf(fact_4_process__term_Osimps_I21_J,axiom,
! [A: $tType,D: $tType,Pvar: $tType,F1: Pvar > D,F2: ( proces634752977rocess @ A ) > D,F3: A > ( proces554692349s_term @ A @ Pvar ) > D > D,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D,X1: Pvar] :
( ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( F1 @ X1 ) ) ).
% process_term.simps(21)
thf(fact_5_process__term_Osimps_I17_J,axiom,
! [A: $tType,B: $tType,Pvar: $tType,F1: Pvar > B,F2: ( proces634752977rocess @ A ) > B,F3: A > ( proces554692349s_term @ A @ Pvar ) > B,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B,X1: Pvar] :
( ( proces460752237s_term @ Pvar @ B @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( F1 @ X1 ) ) ).
% process_term.simps(17)
thf(fact_6_process__term_Oset__intros_I6_J,axiom,
! [A: $tType,Pvar: $tType,Ab: Pvar] : ( member @ Pvar @ Ab @ ( proces1493547885s_term @ A @ Pvar @ ( proces1627516585ss_VAR @ Pvar @ A @ Ab ) ) ) ).
% process_term.set_intros(6)
thf(fact_7_process__term_Osimps_I25_J,axiom,
! [A: $tType,B: $tType,C: $tType,Pvar: $tType,F1: A > B,F2: Pvar > C,X1: Pvar] :
( ( proces224683037s_term @ A @ B @ Pvar @ C @ F1 @ F2 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( proces1627516585ss_VAR @ C @ B @ ( F2 @ X1 ) ) ) ).
% process_term.simps(25)
thf(fact_8_isACT_Osimps_I3_J,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,Sys2: A > ( proces554692349s_term @ B @ C ),A2: D,T2: proces554692349s_term @ D @ A] : ( proces687458811_isACT @ A @ B @ C @ D @ Sys2 @ ( proces1454156180ss_ACT @ D @ A @ A2 @ T2 ) ) ).
% isACT.simps(3)
thf(fact_9_isACT_Osimps_I4_J,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,Sys2: A > ( proces554692349s_term @ B @ C ),T1: proces554692349s_term @ D @ A,T22: proces554692349s_term @ D @ A] :
~ ( proces687458811_isACT @ A @ B @ C @ D @ Sys2 @ ( proces89589571ess_CH @ D @ A @ T1 @ T22 ) ) ).
% isACT.simps(4)
thf(fact_10_process__term_Odistinct_I3_J,axiom,
! [Pvar: $tType,A: $tType,X1: Pvar,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces1627516585ss_VAR @ Pvar @ A @ X1 )
!= ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) ) ).
% process_term.distinct(3)
thf(fact_11_process__term_Odistinct_I5_J,axiom,
! [Pvar: $tType,A: $tType,X1: Pvar,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces1627516585ss_VAR @ Pvar @ A @ X1 )
!= ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) ) ).
% process_term.distinct(5)
thf(fact_12_process__term_Odistinct_I1_J,axiom,
! [Pvar: $tType,A: $tType,X1: Pvar,X2: proces634752977rocess @ A] :
( ( proces1627516585ss_VAR @ Pvar @ A @ X1 )
!= ( proces1062592052s_PROC @ A @ Pvar @ X2 ) ) ).
% process_term.distinct(1)
thf(fact_13_process__term_Orel__inject_I1_J,axiom,
! [B: $tType,A: $tType,Pvar: $tType,C: $tType,R1: A > B > $o,R2: Pvar > C > $o,X1: Pvar,Y1: C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces1627516585ss_VAR @ C @ B @ Y1 ) )
= ( R2 @ X1 @ Y1 ) ) ).
% process_term.rel_inject(1)
thf(fact_14_process__term_Oinject_I2_J,axiom,
! [Pvar: $tType,A: $tType,X2: proces634752977rocess @ A,Y2: proces634752977rocess @ A] :
( ( ( proces1062592052s_PROC @ A @ Pvar @ X2 )
= ( proces1062592052s_PROC @ A @ Pvar @ Y2 ) )
= ( X2 = Y2 ) ) ).
% process_term.inject(2)
thf(fact_15_process__term_Oinject_I4_J,axiom,
! [Pvar: $tType,A: $tType,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar,Y41: proces554692349s_term @ A @ Pvar,Y42: proces554692349s_term @ A @ Pvar] :
( ( ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 )
= ( proces89589571ess_CH @ A @ Pvar @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% process_term.inject(4)
thf(fact_16_process__term_Oinject_I3_J,axiom,
! [Pvar: $tType,A: $tType,X31: A,X32: proces554692349s_term @ A @ Pvar,Y31: A,Y32: proces554692349s_term @ A @ Pvar] :
( ( ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 )
= ( proces1454156180ss_ACT @ A @ Pvar @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% process_term.inject(3)
thf(fact_17_process__term_Opred__inject_I4_J,axiom,
! [Pvar: $tType,A: $tType,P1: A > $o,P2: Pvar > $o,A2: proces554692349s_term @ A @ Pvar,Aa2: proces554692349s_term @ A @ Pvar] :
( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ ( proces89589571ess_CH @ A @ Pvar @ A2 @ Aa2 ) )
= ( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ A2 )
& ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ Aa2 ) ) ) ).
% process_term.pred_inject(4)
thf(fact_18_process__term_Opred__inject_I3_J,axiom,
! [Pvar: $tType,A: $tType,P1: A > $o,P2: Pvar > $o,A2: A,Aa2: proces554692349s_term @ A @ Pvar] :
( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ ( proces1454156180ss_ACT @ A @ Pvar @ A2 @ Aa2 ) )
= ( ( P1 @ A2 )
& ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ Aa2 ) ) ) ).
% process_term.pred_inject(3)
thf(fact_19_process__term_Osimps_I51_J,axiom,
! [Pvar: $tType,A: $tType,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces1493547885s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( proces1493547885s_term @ A @ Pvar @ X32 ) ) ).
% process_term.simps(51)
thf(fact_20_process__term_Osimps_I28_J,axiom,
! [C: $tType,B: $tType,Pvar: $tType,A: $tType,F1: A > B,F2: Pvar > C,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces224683037s_term @ A @ B @ Pvar @ C @ F1 @ F2 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) )
= ( proces89589571ess_CH @ B @ C @ ( proces224683037s_term @ A @ B @ Pvar @ C @ F1 @ F2 @ X41 ) @ ( proces224683037s_term @ A @ B @ Pvar @ C @ F1 @ F2 @ X42 ) ) ) ).
% process_term.simps(28)
thf(fact_21_process__term_Osimps_I27_J,axiom,
! [C: $tType,B: $tType,Pvar: $tType,A: $tType,F1: A > B,F2: Pvar > C,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces224683037s_term @ A @ B @ Pvar @ C @ F1 @ F2 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( proces1454156180ss_ACT @ B @ C @ ( F1 @ X31 ) @ ( proces224683037s_term @ A @ B @ Pvar @ C @ F1 @ F2 @ X32 ) ) ) ).
% process_term.simps(27)
thf(fact_22_process__term_Osimps_I24_J,axiom,
! [D: $tType,Pvar: $tType,A: $tType,F1: Pvar > D,F2: ( proces634752977rocess @ A ) > D,F3: A > ( proces554692349s_term @ A @ Pvar ) > D > D,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) )
= ( F4 @ X41 @ X42 @ ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ X41 ) @ ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ X42 ) ) ) ).
% process_term.simps(24)
thf(fact_23_process__term_Osimps_I23_J,axiom,
! [D: $tType,Pvar: $tType,A: $tType,F1: Pvar > D,F2: ( proces634752977rocess @ A ) > D,F3: A > ( proces554692349s_term @ A @ Pvar ) > D > D,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( F3 @ X31 @ X32 @ ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ X32 ) ) ) ).
% process_term.simps(23)
thf(fact_24_process__term_Osimps_I22_J,axiom,
! [Pvar: $tType,D: $tType,A: $tType,F1: Pvar > D,F2: ( proces634752977rocess @ A ) > D,F3: A > ( proces554692349s_term @ A @ Pvar ) > D > D,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D,X2: proces634752977rocess @ A] :
( ( proces2117273769s_term @ Pvar @ D @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1062592052s_PROC @ A @ Pvar @ X2 ) )
= ( F2 @ X2 ) ) ).
% process_term.simps(22)
thf(fact_25_process__term_Osimps_I20_J,axiom,
! [B: $tType,Pvar: $tType,A: $tType,F1: Pvar > B,F2: ( proces634752977rocess @ A ) > B,F3: A > ( proces554692349s_term @ A @ Pvar ) > B,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces460752237s_term @ Pvar @ B @ A @ F1 @ F2 @ F3 @ F4 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) )
= ( F4 @ X41 @ X42 ) ) ).
% process_term.simps(20)
thf(fact_26_process__term_Osimps_I19_J,axiom,
! [B: $tType,Pvar: $tType,A: $tType,F1: Pvar > B,F2: ( proces634752977rocess @ A ) > B,F3: A > ( proces554692349s_term @ A @ Pvar ) > B,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces460752237s_term @ Pvar @ B @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( F3 @ X31 @ X32 ) ) ).
% process_term.simps(19)
thf(fact_27_process__term_Osimps_I18_J,axiom,
! [Pvar: $tType,B: $tType,A: $tType,F1: Pvar > B,F2: ( proces634752977rocess @ A ) > B,F3: A > ( proces554692349s_term @ A @ Pvar ) > B,F4: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > B,X2: proces634752977rocess @ A] :
( ( proces460752237s_term @ Pvar @ B @ A @ F1 @ F2 @ F3 @ F4 @ ( proces1062592052s_PROC @ A @ Pvar @ X2 ) )
= ( F2 @ X2 ) ) ).
% process_term.simps(18)
thf(fact_28_process__term_Odistinct_I11_J,axiom,
! [Pvar: $tType,A: $tType,X31: A,X32: proces554692349s_term @ A @ Pvar,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 )
!= ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) ) ).
% process_term.distinct(11)
thf(fact_29_process__term_Odistinct_I9_J,axiom,
! [Pvar: $tType,A: $tType,X2: proces634752977rocess @ A,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar] :
( ( proces1062592052s_PROC @ A @ Pvar @ X2 )
!= ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) ) ).
% process_term.distinct(9)
thf(fact_30_process__term_Odistinct_I7_J,axiom,
! [Pvar: $tType,A: $tType,X2: proces634752977rocess @ A,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces1062592052s_PROC @ A @ Pvar @ X2 )
!= ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) ) ).
% process_term.distinct(7)
thf(fact_31_process__term_Oset__cases_I2_J,axiom,
! [Pvar: $tType,A: $tType,E: Pvar,A2: proces554692349s_term @ A @ Pvar] :
( ( member @ Pvar @ E @ ( proces1493547885s_term @ A @ Pvar @ A2 ) )
=> ( ( A2
!= ( proces1627516585ss_VAR @ Pvar @ A @ E ) )
=> ( ! [Z1: A,Z2: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces1454156180ss_ACT @ A @ Pvar @ Z1 @ Z2 ) )
=> ~ ( member @ Pvar @ E @ ( proces1493547885s_term @ A @ Pvar @ Z2 ) ) )
=> ( ! [Z1: proces554692349s_term @ A @ Pvar] :
( ? [Z2: proces554692349s_term @ A @ Pvar] :
( A2
= ( proces89589571ess_CH @ A @ Pvar @ Z1 @ Z2 ) )
=> ~ ( member @ Pvar @ E @ ( proces1493547885s_term @ A @ Pvar @ Z1 ) ) )
=> ~ ! [Z1: proces554692349s_term @ A @ Pvar,Z2: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces89589571ess_CH @ A @ Pvar @ Z1 @ Z2 ) )
=> ~ ( member @ Pvar @ E @ ( proces1493547885s_term @ A @ Pvar @ Z2 ) ) ) ) ) ) ) ).
% process_term.set_cases(2)
thf(fact_32_process__term_Orel__inject_I4_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X41: proces554692349s_term @ A @ Pvar,X42: proces554692349s_term @ A @ Pvar,Y41: proces554692349s_term @ B @ C,Y42: proces554692349s_term @ B @ C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) )
= ( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X41 @ Y41 )
& ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X42 @ Y42 ) ) ) ).
% process_term.rel_inject(4)
thf(fact_33_process__term_Orel__inject_I3_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X31: A,X32: proces554692349s_term @ A @ Pvar,Y31: B,Y32: proces554692349s_term @ B @ C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) @ ( proces1454156180ss_ACT @ B @ C @ Y31 @ Y32 ) )
= ( ( R1 @ X31 @ Y31 )
& ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X32 @ Y32 ) ) ) ).
% process_term.rel_inject(3)
thf(fact_34_process__term_Orel__intros_I4_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X41: proces554692349s_term @ A @ Pvar,Y41: proces554692349s_term @ B @ C,X42: proces554692349s_term @ A @ Pvar,Y42: proces554692349s_term @ B @ C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X41 @ Y41 )
=> ( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X42 @ Y42 )
=> ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces89589571ess_CH @ A @ Pvar @ X41 @ X42 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) ) ) ) ).
% process_term.rel_intros(4)
thf(fact_35_process__term_Orel__intros_I3_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,X31: A,Y31: B,R2: Pvar > C > $o,X32: proces554692349s_term @ A @ Pvar,Y32: proces554692349s_term @ B @ C] :
( ( R1 @ X31 @ Y31 )
=> ( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X32 @ Y32 )
=> ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) @ ( proces1454156180ss_ACT @ B @ C @ Y31 @ Y32 ) ) ) ) ).
% process_term.rel_intros(3)
thf(fact_36_process__term_Oset__intros_I9_J,axiom,
! [Pvar: $tType,A: $tType,Xm: Pvar,A2c: proces554692349s_term @ A @ Pvar,A1c: proces554692349s_term @ A @ Pvar] :
( ( member @ Pvar @ Xm @ ( proces1493547885s_term @ A @ Pvar @ A2c ) )
=> ( member @ Pvar @ Xm @ ( proces1493547885s_term @ A @ Pvar @ ( proces89589571ess_CH @ A @ Pvar @ A1c @ A2c ) ) ) ) ).
% process_term.set_intros(9)
thf(fact_37_process__term_Oset__intros_I8_J,axiom,
! [Pvar: $tType,A: $tType,Xk: Pvar,A1c: proces554692349s_term @ A @ Pvar,A2c: proces554692349s_term @ A @ Pvar] :
( ( member @ Pvar @ Xk @ ( proces1493547885s_term @ A @ Pvar @ A1c ) )
=> ( member @ Pvar @ Xk @ ( proces1493547885s_term @ A @ Pvar @ ( proces89589571ess_CH @ A @ Pvar @ A1c @ A2c ) ) ) ) ).
% process_term.set_intros(8)
thf(fact_38_process__term_Oset__intros_I7_J,axiom,
! [Pvar: $tType,A: $tType,Xi: Pvar,A2b: proces554692349s_term @ A @ Pvar,A1b: A] :
( ( member @ Pvar @ Xi @ ( proces1493547885s_term @ A @ Pvar @ A2b ) )
=> ( member @ Pvar @ Xi @ ( proces1493547885s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ A1b @ A2b ) ) ) ) ).
% process_term.set_intros(7)
thf(fact_39_process__term_Orel__distinct_I12_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,Y41: proces554692349s_term @ A @ B,Y42: proces554692349s_term @ A @ B,X31: C,X32: proces554692349s_term @ C @ D] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 @ ( proces89589571ess_CH @ A @ B @ Y41 @ Y42 ) @ ( proces1454156180ss_ACT @ C @ D @ X31 @ X32 ) ) ).
% process_term.rel_distinct(12)
thf(fact_40_process__term_Orel__distinct_I11_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X31: A,X32: proces554692349s_term @ A @ Pvar,Y41: proces554692349s_term @ B @ C,Y42: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) ) ).
% process_term.rel_distinct(11)
thf(fact_41_process__term_Orel__distinct_I10_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,Y41: proces554692349s_term @ A @ B,Y42: proces554692349s_term @ A @ B,X2: proces634752977rocess @ C] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 @ ( proces89589571ess_CH @ A @ B @ Y41 @ Y42 ) @ ( proces1062592052s_PROC @ C @ D @ X2 ) ) ).
% process_term.rel_distinct(10)
thf(fact_42_process__term_Orel__distinct_I9_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X2: proces634752977rocess @ A,Y41: proces554692349s_term @ B @ C,Y42: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1062592052s_PROC @ A @ Pvar @ X2 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) ) ).
% process_term.rel_distinct(9)
thf(fact_43_process__term_Orel__distinct_I8_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,Y31: A,Y32: proces554692349s_term @ A @ B,X2: proces634752977rocess @ C] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 @ ( proces1454156180ss_ACT @ A @ B @ Y31 @ Y32 ) @ ( proces1062592052s_PROC @ C @ D @ X2 ) ) ).
% process_term.rel_distinct(8)
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_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_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_process__term_Orel__distinct_I7_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X2: proces634752977rocess @ A,Y31: B,Y32: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1062592052s_PROC @ A @ Pvar @ X2 ) @ ( proces1454156180ss_ACT @ B @ C @ Y31 @ Y32 ) ) ).
% process_term.rel_distinct(7)
thf(fact_49_CH2_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > ( proces554692349s_term @ B @ A ) ) > ( proces554692349s_term @ B @ A ) > $o,A0: A > ( proces554692349s_term @ B @ A ),A1: proces554692349s_term @ B @ A] :
( ! [Sys3: A > ( proces554692349s_term @ B @ A ),X5: A] : ( P @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X5 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),P3: proces634752977rocess @ B] : ( P @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P3 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),T12: proces554692349s_term @ B @ A,T23: proces554692349s_term @ B @ A] : ( P @ Sys3 @ ( proces89589571ess_CH @ B @ A @ T12 @ T23 ) )
=> ( ! [A4: A > ( proces554692349s_term @ B @ A ),V: B,Va: proces554692349s_term @ B @ A] : ( P @ A4 @ ( proces1454156180ss_ACT @ B @ A @ V @ Va ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% CH2.induct
thf(fact_50_CONT_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > ( proces554692349s_term @ B @ A ) ) > ( proces554692349s_term @ B @ A ) > $o,A0: A > ( proces554692349s_term @ B @ A ),A1: proces554692349s_term @ B @ A] :
( ! [Sys3: A > ( proces554692349s_term @ B @ A ),X5: A] : ( P @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X5 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),P3: proces634752977rocess @ B] : ( P @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P3 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),A4: B,T3: proces554692349s_term @ B @ A] : ( P @ Sys3 @ ( proces1454156180ss_ACT @ B @ A @ A4 @ T3 ) )
=> ( ! [A4: A > ( proces554692349s_term @ B @ A ),V: proces554692349s_term @ B @ A,Va: proces554692349s_term @ B @ A] : ( P @ A4 @ ( proces89589571ess_CH @ B @ A @ V @ Va ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% CONT.induct
thf(fact_51_PREF_Oinduct,axiom,
! [C: $tType,A: $tType,B: $tType,P: ( A > ( proces554692349s_term @ B @ C ) ) > ( proces554692349s_term @ B @ A ) > $o,A0: A > ( proces554692349s_term @ B @ C ),A1: proces554692349s_term @ B @ A] :
( ! [Sys3: A > ( proces554692349s_term @ B @ C ),X5: A] : ( P @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X5 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),P3: proces634752977rocess @ B] : ( P @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P3 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),A4: B,T3: proces554692349s_term @ B @ A] : ( P @ Sys3 @ ( proces1454156180ss_ACT @ B @ A @ A4 @ T3 ) )
=> ( ! [A4: A > ( proces554692349s_term @ B @ C ),V: proces554692349s_term @ B @ A,Va: proces554692349s_term @ B @ A] : ( P @ A4 @ ( proces89589571ess_CH @ B @ A @ V @ Va ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% PREF.induct
thf(fact_52_isACT_Oinduct,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,P: ( A > ( proces554692349s_term @ B @ C ) ) > ( proces554692349s_term @ D @ A ) > $o,A0: A > ( proces554692349s_term @ B @ C ),A1: proces554692349s_term @ D @ A] :
( ! [Sys3: A > ( proces554692349s_term @ B @ C ),X5: A] : ( P @ Sys3 @ ( proces1627516585ss_VAR @ A @ D @ X5 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),P3: proces634752977rocess @ D] : ( P @ Sys3 @ ( proces1062592052s_PROC @ D @ A @ P3 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),A4: D,T3: proces554692349s_term @ D @ A] : ( P @ Sys3 @ ( proces1454156180ss_ACT @ D @ A @ A4 @ T3 ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),T12: proces554692349s_term @ D @ A,T23: proces554692349s_term @ D @ A] : ( P @ Sys3 @ ( proces89589571ess_CH @ D @ A @ T12 @ T23 ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% isACT.induct
thf(fact_53_process__term_Oinduct,axiom,
! [Pvar: $tType,A: $tType,P: ( proces554692349s_term @ A @ Pvar ) > $o,Process_term: proces554692349s_term @ A @ Pvar] :
( ! [X4: Pvar] : ( P @ ( proces1627516585ss_VAR @ Pvar @ A @ X4 ) )
=> ( ! [X4: proces634752977rocess @ A] : ( P @ ( proces1062592052s_PROC @ A @ Pvar @ X4 ) )
=> ( ! [X1a: A,X2a: proces554692349s_term @ A @ Pvar] :
( ( P @ X2a )
=> ( P @ ( proces1454156180ss_ACT @ A @ Pvar @ X1a @ X2a ) ) )
=> ( ! [X1a: proces554692349s_term @ A @ Pvar,X2a: proces554692349s_term @ A @ Pvar] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( proces89589571ess_CH @ A @ Pvar @ X1a @ X2a ) ) ) )
=> ( P @ Process_term ) ) ) ) ) ).
% process_term.induct
thf(fact_54_process__term_Oexhaust,axiom,
! [Pvar: $tType,A: $tType,Y3: proces554692349s_term @ A @ Pvar] :
( ! [X12: Pvar] :
( Y3
!= ( proces1627516585ss_VAR @ Pvar @ A @ X12 ) )
=> ( ! [X22: proces634752977rocess @ A] :
( Y3
!= ( proces1062592052s_PROC @ A @ Pvar @ X22 ) )
=> ( ! [X312: A,X322: proces554692349s_term @ A @ Pvar] :
( Y3
!= ( proces1454156180ss_ACT @ A @ Pvar @ X312 @ X322 ) )
=> ~ ! [X412: proces554692349s_term @ A @ Pvar,X422: proces554692349s_term @ A @ Pvar] :
( Y3
!= ( proces89589571ess_CH @ A @ Pvar @ X412 @ X422 ) ) ) ) ) ).
% process_term.exhaust
thf(fact_55_process__term_Orel__eq,axiom,
! [B: $tType,A: $tType] :
( ( proces2029722208s_term @ A @ A @ B @ B
@ ^ [Y4: A,Z: A] : Y4 = Z
@ ^ [Y4: B,Z: B] : Y4 = Z )
= ( ^ [Y4: proces554692349s_term @ A @ B,Z: proces554692349s_term @ A @ B] : Y4 = Z ) ) ).
% process_term.rel_eq
thf(fact_56_process__term_Orel__refl,axiom,
! [D: $tType,C: $tType,R1a: C > C > $o,R2a: D > D > $o,X6: proces554692349s_term @ C @ D] :
( ! [X4: C] : ( R1a @ X4 @ X4 )
=> ( ! [X4: D] : ( R2a @ X4 @ X4 )
=> ( proces2029722208s_term @ C @ C @ D @ D @ R1a @ R2a @ X6 @ X6 ) ) ) ).
% process_term.rel_refl
thf(fact_57_process__term_Orel__distinct_I1_J,axiom,
! [Pvar: $tType,A: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X1: Pvar,Y2: proces634752977rocess @ B] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces1062592052s_PROC @ B @ C @ Y2 ) ) ).
% process_term.rel_distinct(1)
thf(fact_58_process__term_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > C > $o,R2: B > D > $o,Y2: proces634752977rocess @ A,X1: D] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 @ ( proces1062592052s_PROC @ A @ B @ Y2 ) @ ( proces1627516585ss_VAR @ D @ C @ X1 ) ) ).
% process_term.rel_distinct(2)
thf(fact_59_process__term_Orel__distinct_I5_J,axiom,
! [Pvar: $tType,A: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X1: Pvar,Y41: proces554692349s_term @ B @ C,Y42: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces89589571ess_CH @ B @ C @ Y41 @ Y42 ) ) ).
% process_term.rel_distinct(5)
thf(fact_60_process__term_Orel__distinct_I6_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > C > $o,R2: B > D > $o,Y41: proces554692349s_term @ A @ B,Y42: proces554692349s_term @ A @ B,X1: D] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 @ ( proces89589571ess_CH @ A @ B @ Y41 @ Y42 ) @ ( proces1627516585ss_VAR @ D @ C @ X1 ) ) ).
% process_term.rel_distinct(6)
thf(fact_61_process__term_Orel__distinct_I3_J,axiom,
! [Pvar: $tType,A: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X1: Pvar,Y31: B,Y32: proces554692349s_term @ B @ C] :
~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces1454156180ss_ACT @ B @ C @ Y31 @ Y32 ) ) ).
% process_term.rel_distinct(3)
thf(fact_62_process__term_Orel__distinct_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > C > $o,R2: B > D > $o,Y31: A,Y32: proces554692349s_term @ A @ B,X1: D] :
~ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 @ ( proces1454156180ss_ACT @ A @ B @ Y31 @ Y32 ) @ ( proces1627516585ss_VAR @ D @ C @ X1 ) ) ).
% process_term.rel_distinct(4)
thf(fact_63_process__term_Orel__intros_I1_J,axiom,
! [Pvar: $tType,A: $tType,B: $tType,C: $tType,R2: Pvar > C > $o,X1: Pvar,Y1: C,R1: A > B > $o] :
( ( R2 @ X1 @ Y1 )
=> ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) @ ( proces1627516585ss_VAR @ C @ B @ Y1 ) ) ) ).
% process_term.rel_intros(1)
thf(fact_64_process__term_Orel__cases,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,A2: proces554692349s_term @ A @ Pvar,B2: proces554692349s_term @ B @ C] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ A2 @ B2 )
=> ( ! [X4: Pvar] :
( ( A2
= ( proces1627516585ss_VAR @ Pvar @ A @ X4 ) )
=> ! [Y5: C] :
( ( B2
= ( proces1627516585ss_VAR @ C @ B @ Y5 ) )
=> ~ ( R2 @ X4 @ Y5 ) ) )
=> ( ! [Xa: proces634752977rocess @ A] :
( ( A2
= ( proces1062592052s_PROC @ A @ Pvar @ Xa ) )
=> ! [Ya: proces634752977rocess @ B] :
( ( B2
= ( proces1062592052s_PROC @ B @ C @ Ya ) )
=> ~ ( proces749077512rocess @ A @ B @ R1 @ Xa @ Ya ) ) )
=> ( ! [X1a: A,X2a: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces1454156180ss_ACT @ A @ Pvar @ X1a @ X2a ) )
=> ! [Y1a: B,Y2a: proces554692349s_term @ B @ C] :
( ( B2
= ( proces1454156180ss_ACT @ B @ C @ Y1a @ Y2a ) )
=> ( ( R1 @ X1a @ Y1a )
=> ~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X2a @ Y2a ) ) ) )
=> ~ ! [X1b: proces554692349s_term @ A @ Pvar,X2b: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces89589571ess_CH @ A @ Pvar @ X1b @ X2b ) )
=> ! [Y1b: proces554692349s_term @ B @ C,Y2b: proces554692349s_term @ B @ C] :
( ( B2
= ( proces89589571ess_CH @ B @ C @ Y1b @ Y2b ) )
=> ( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X1b @ Y1b )
=> ~ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X2b @ Y2b ) ) ) ) ) ) ) ) ).
% process_term.rel_cases
thf(fact_65_process__term_Orel__induct,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X6: proces554692349s_term @ A @ Pvar,Y3: proces554692349s_term @ B @ C,Q: ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ B @ C ) > $o] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ X6 @ Y3 )
=> ( ! [A12: Pvar,B1: C] :
( ( R2 @ A12 @ B1 )
=> ( Q @ ( proces1627516585ss_VAR @ Pvar @ A @ A12 ) @ ( proces1627516585ss_VAR @ C @ B @ B1 ) ) )
=> ( ! [A22: proces634752977rocess @ A,B22: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R1 @ A22 @ B22 )
=> ( Q @ ( proces1062592052s_PROC @ A @ Pvar @ A22 ) @ ( proces1062592052s_PROC @ B @ C @ B22 ) ) )
=> ( ! [A31: A,A32: proces554692349s_term @ A @ Pvar,B31: B,B32: proces554692349s_term @ B @ C] :
( ( R1 @ A31 @ B31 )
=> ( ( Q @ A32 @ B32 )
=> ( Q @ ( proces1454156180ss_ACT @ A @ Pvar @ A31 @ A32 ) @ ( proces1454156180ss_ACT @ B @ C @ B31 @ B32 ) ) ) )
=> ( ! [A41: proces554692349s_term @ A @ Pvar,A42: proces554692349s_term @ A @ Pvar,B41: proces554692349s_term @ B @ C,B42: proces554692349s_term @ B @ C] :
( ( Q @ A41 @ B41 )
=> ( ( Q @ A42 @ B42 )
=> ( Q @ ( proces89589571ess_CH @ A @ Pvar @ A41 @ A42 ) @ ( proces89589571ess_CH @ B @ C @ B41 @ B42 ) ) ) )
=> ( Q @ X6 @ Y3 ) ) ) ) ) ) ).
% process_term.rel_induct
thf(fact_66_CH2_Ocases,axiom,
! [A: $tType,B: $tType,X6: product_prod @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A )] :
( ! [Sys3: A > ( proces554692349s_term @ B @ A ),X5: A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X5 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),P3: proces634752977rocess @ B] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P3 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),T12: proces554692349s_term @ B @ A,T23: proces554692349s_term @ B @ A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces89589571ess_CH @ B @ A @ T12 @ T23 ) ) )
=> ~ ! [A4: A > ( proces554692349s_term @ B @ A ),V: B,Va: proces554692349s_term @ B @ A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ A4 @ ( proces1454156180ss_ACT @ B @ A @ V @ Va ) ) ) ) ) ) ).
% CH2.cases
thf(fact_67_CONT_Ocases,axiom,
! [A: $tType,B: $tType,X6: product_prod @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A )] :
( ! [Sys3: A > ( proces554692349s_term @ B @ A ),X5: A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X5 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),P3: proces634752977rocess @ B] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P3 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ A ),A4: B,T3: proces554692349s_term @ B @ A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1454156180ss_ACT @ B @ A @ A4 @ T3 ) ) )
=> ~ ! [A4: A > ( proces554692349s_term @ B @ A ),V: proces554692349s_term @ B @ A,Va: proces554692349s_term @ B @ A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ A ) ) @ ( proces554692349s_term @ B @ A ) @ A4 @ ( proces89589571ess_CH @ B @ A @ V @ Va ) ) ) ) ) ) ).
% CONT.cases
thf(fact_68_PREF_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X6: product_prod @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A )] :
( ! [Sys3: A > ( proces554692349s_term @ B @ C ),X5: A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1627516585ss_VAR @ A @ B @ X5 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),P3: proces634752977rocess @ B] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1062592052s_PROC @ B @ A @ P3 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),A4: B,T3: proces554692349s_term @ B @ A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A ) @ Sys3 @ ( proces1454156180ss_ACT @ B @ A @ A4 @ T3 ) ) )
=> ~ ! [A4: A > ( proces554692349s_term @ B @ C ),V: proces554692349s_term @ B @ A,Va: proces554692349s_term @ B @ A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ B @ A ) @ A4 @ ( proces89589571ess_CH @ B @ A @ V @ Va ) ) ) ) ) ) ).
% PREF.cases
thf(fact_69_isACT_Ocases,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,X6: product_prod @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A )] :
( ! [Sys3: A > ( proces554692349s_term @ B @ C ),X5: A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A ) @ Sys3 @ ( proces1627516585ss_VAR @ A @ D @ X5 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),P3: proces634752977rocess @ D] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A ) @ Sys3 @ ( proces1062592052s_PROC @ D @ A @ P3 ) ) )
=> ( ! [Sys3: A > ( proces554692349s_term @ B @ C ),A4: D,T3: proces554692349s_term @ D @ A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A ) @ Sys3 @ ( proces1454156180ss_ACT @ D @ A @ A4 @ T3 ) ) )
=> ~ ! [Sys3: A > ( proces554692349s_term @ B @ C ),T12: proces554692349s_term @ D @ A,T23: proces554692349s_term @ D @ A] :
( X6
!= ( product_Pair @ ( A > ( proces554692349s_term @ B @ C ) ) @ ( proces554692349s_term @ D @ A ) @ Sys3 @ ( proces89589571ess_CH @ D @ A @ T12 @ T23 ) ) ) ) ) ) ).
% isACT.cases
thf(fact_70_process__term_Opred__inject_I2_J,axiom,
! [Pvar: $tType,A: $tType,P1: A > $o,P2: Pvar > $o,A2: proces634752977rocess @ A] :
( ( proces2118920028s_term @ A @ Pvar @ P1 @ P2 @ ( proces1062592052s_PROC @ A @ Pvar @ A2 ) )
= ( proces1205983068rocess @ A @ P1 @ A2 ) ) ).
% process_term.pred_inject(2)
thf(fact_71_process__term_Opred__cong,axiom,
! [B: $tType,A: $tType,X6: proces554692349s_term @ A @ B,Ya2: proces554692349s_term @ A @ B,P1: A > $o,P1a: A > $o,P2: B > $o,P2a: B > $o] :
( ( X6 = Ya2 )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ Ya2 ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( proces1493547885s_term @ A @ B @ Ya2 ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( proces2118920028s_term @ A @ B @ P1 @ P2 @ X6 )
= ( proces2118920028s_term @ A @ B @ P1a @ P2a @ Ya2 ) ) ) ) ) ).
% process_term.pred_cong
thf(fact_72_process__term_Opred__mono__strong,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,X6: proces554692349s_term @ A @ B,P1a: A > $o,P2a: B > $o] :
( ( proces2118920028s_term @ A @ B @ P1 @ P2 @ X6 )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ X6 ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( proces1493547885s_term @ A @ B @ X6 ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( proces2118920028s_term @ A @ B @ P1a @ P2a @ X6 ) ) ) ) ).
% process_term.pred_mono_strong
thf(fact_73_process__term_Omap__cong,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X6: proces554692349s_term @ A @ B,Ya2: proces554692349s_term @ A @ B,F1: A > C,G1: A > C,F2: B > D,G2: B > D] :
( ( X6 = Ya2 )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ Ya2 ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( proces1493547885s_term @ A @ B @ Ya2 ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( proces224683037s_term @ A @ C @ B @ D @ F1 @ F2 @ X6 )
= ( proces224683037s_term @ A @ C @ B @ D @ G1 @ G2 @ Ya2 ) ) ) ) ) ).
% process_term.map_cong
thf(fact_74_process_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X6: proces634752977rocess @ B] :
( ! [X4: B] : ( Ra @ X4 @ X4 )
=> ( proces749077512rocess @ B @ B @ Ra @ X6 @ X6 ) ) ).
% process.rel_refl
thf(fact_75_process_Orel__eq,axiom,
! [A: $tType] :
( ( proces749077512rocess @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [Y4: proces634752977rocess @ A,Z: proces634752977rocess @ A] : Y4 = Z ) ) ).
% process.rel_eq
thf(fact_76_process__term_Oset__intros_I4_J,axiom,
! [Pvar: $tType,A: $tType,Xc: A,A1a: proces554692349s_term @ A @ Pvar,A2a: proces554692349s_term @ A @ Pvar] :
( ( member @ A @ Xc @ ( proces1264666542s_term @ A @ Pvar @ A1a ) )
=> ( member @ A @ Xc @ ( proces1264666542s_term @ A @ Pvar @ ( proces89589571ess_CH @ A @ Pvar @ A1a @ A2a ) ) ) ) ).
% process_term.set_intros(4)
thf(fact_77_process__term_Oset__intros_I5_J,axiom,
! [Pvar: $tType,A: $tType,Xe: A,A2a: proces554692349s_term @ A @ Pvar,A1a: proces554692349s_term @ A @ Pvar] :
( ( member @ A @ Xe @ ( proces1264666542s_term @ A @ Pvar @ A2a ) )
=> ( member @ A @ Xe @ ( proces1264666542s_term @ A @ Pvar @ ( proces89589571ess_CH @ A @ Pvar @ A1a @ A2a ) ) ) ) ).
% process_term.set_intros(5)
thf(fact_78_process__term_Oset__intros_I2_J,axiom,
! [Pvar: $tType,A: $tType,A1: A,A23: proces554692349s_term @ A @ Pvar] : ( member @ A @ A1 @ ( proces1264666542s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ A1 @ A23 ) ) ) ).
% process_term.set_intros(2)
thf(fact_79_process__term_Oset__intros_I3_J,axiom,
! [Pvar: $tType,A: $tType,Xa2: A,A23: proces554692349s_term @ A @ Pvar,A1: A] :
( ( member @ A @ Xa2 @ ( proces1264666542s_term @ A @ Pvar @ A23 ) )
=> ( member @ A @ Xa2 @ ( proces1264666542s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ A1 @ A23 ) ) ) ) ).
% process_term.set_intros(3)
thf(fact_80_process__term_Orel__intros_I2_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,X2: proces634752977rocess @ A,Y2: proces634752977rocess @ B,R2: Pvar > C > $o] :
( ( proces749077512rocess @ A @ B @ R1 @ X2 @ Y2 )
=> ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1062592052s_PROC @ A @ Pvar @ X2 ) @ ( proces1062592052s_PROC @ B @ C @ Y2 ) ) ) ).
% process_term.rel_intros(2)
thf(fact_81_process__term_Orel__inject_I2_J,axiom,
! [C: $tType,Pvar: $tType,A: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o,X2: proces634752977rocess @ A,Y2: proces634752977rocess @ B] :
( ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 @ ( proces1062592052s_PROC @ A @ Pvar @ X2 ) @ ( proces1062592052s_PROC @ B @ C @ Y2 ) )
= ( proces749077512rocess @ A @ B @ R1 @ X2 @ Y2 ) ) ).
% process_term.rel_inject(2)
thf(fact_82_process__term_Orel__refl__strong,axiom,
! [B: $tType,A: $tType,X6: proces554692349s_term @ A @ B,R1a: A > A > $o,R2a: B > B > $o] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ X6 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( proces1493547885s_term @ A @ B @ X6 ) )
=> ( R2a @ Z2 @ Z2 ) )
=> ( proces2029722208s_term @ A @ A @ B @ B @ R1a @ R2a @ X6 @ X6 ) ) ) ).
% process_term.rel_refl_strong
thf(fact_83_process__term_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,X6: proces554692349s_term @ A @ B,Y3: proces554692349s_term @ C @ D,R1a: A > C > $o,R2a: B > D > $o] :
( ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 @ X6 @ Y3 )
=> ( ! [Z1: A,Y12: C] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ X6 ) )
=> ( ( member @ C @ Y12 @ ( proces1264666542s_term @ C @ D @ Y3 ) )
=> ( ( R1 @ Z1 @ Y12 )
=> ( R1a @ Z1 @ Y12 ) ) ) )
=> ( ! [Z2: B,Y22: D] :
( ( member @ B @ Z2 @ ( proces1493547885s_term @ A @ B @ X6 ) )
=> ( ( member @ D @ Y22 @ ( proces1493547885s_term @ C @ D @ Y3 ) )
=> ( ( R2 @ Z2 @ Y22 )
=> ( R2a @ Z2 @ Y22 ) ) ) )
=> ( proces2029722208s_term @ A @ C @ B @ D @ R1a @ R2a @ X6 @ Y3 ) ) ) ) ).
% process_term.rel_mono_strong
thf(fact_84_process__term_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,X6: proces554692349s_term @ A @ B,Ya2: proces554692349s_term @ A @ B,Y3: proces554692349s_term @ C @ D,Xa2: proces554692349s_term @ C @ D,R1: A > C > $o,R1a: A > C > $o,R2: B > D > $o,R2a: B > D > $o] :
( ( X6 = Ya2 )
=> ( ( Y3 = Xa2 )
=> ( ! [Z1: A,Y12: C] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ Ya2 ) )
=> ( ( member @ C @ Y12 @ ( proces1264666542s_term @ C @ D @ Xa2 ) )
=> ( ( R1 @ Z1 @ Y12 )
= ( R1a @ Z1 @ Y12 ) ) ) )
=> ( ! [Z2: B,Y22: D] :
( ( member @ B @ Z2 @ ( proces1493547885s_term @ A @ B @ Ya2 ) )
=> ( ( member @ D @ Y22 @ ( proces1493547885s_term @ C @ D @ Xa2 ) )
=> ( ( R2 @ Z2 @ Y22 )
= ( R2a @ Z2 @ Y22 ) ) ) )
=> ( ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 @ X6 @ Y3 )
= ( proces2029722208s_term @ A @ C @ B @ D @ R1a @ R2a @ Ya2 @ Xa2 ) ) ) ) ) ) ).
% process_term.rel_cong
thf(fact_85_process__term_Oinj__map__strong,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,X6: proces554692349s_term @ A @ B,Xa2: proces554692349s_term @ A @ B,F1: A > C,F1a: A > C,F2: B > D,F2a: B > D] :
( ! [Z1: A,Z1a: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ X6 ) )
=> ( ( member @ A @ Z1a @ ( proces1264666542s_term @ A @ B @ Xa2 ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: B,Z2a: B] :
( ( member @ B @ Z2 @ ( proces1493547885s_term @ A @ B @ X6 ) )
=> ( ( member @ B @ Z2a @ ( proces1493547885s_term @ A @ B @ Xa2 ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( proces224683037s_term @ A @ C @ B @ D @ F1 @ F2 @ X6 )
= ( proces224683037s_term @ A @ C @ B @ D @ F1a @ F2a @ Xa2 ) )
=> ( X6 = Xa2 ) ) ) ) ).
% process_term.inj_map_strong
thf(fact_86_process__term_Omap__cong0,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X6: proces554692349s_term @ A @ B,F1: A > C,G1: A > C,F2: B > D,G2: B > D] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( proces1264666542s_term @ A @ B @ X6 ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( proces1493547885s_term @ A @ B @ X6 ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( proces224683037s_term @ A @ C @ B @ D @ F1 @ F2 @ X6 )
= ( proces224683037s_term @ A @ C @ B @ D @ G1 @ G2 @ X6 ) ) ) ) ).
% process_term.map_cong0
thf(fact_87_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B3 ) )
= ( ( A2 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_88_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_89_process__term_Oset__cases_I1_J,axiom,
! [Pvar: $tType,A: $tType,E: A,A2: proces554692349s_term @ A @ Pvar] :
( ( member @ A @ E @ ( proces1264666542s_term @ A @ Pvar @ A2 ) )
=> ( ! [Z3: proces634752977rocess @ A] :
( ( A2
= ( proces1062592052s_PROC @ A @ Pvar @ Z3 ) )
=> ~ ( member @ A @ E @ ( proces1148692671rocess @ A @ Z3 ) ) )
=> ( ! [Z2: proces554692349s_term @ A @ Pvar] :
( A2
!= ( proces1454156180ss_ACT @ A @ Pvar @ E @ Z2 ) )
=> ( ! [Z1: A,Z2: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces1454156180ss_ACT @ A @ Pvar @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( proces1264666542s_term @ A @ Pvar @ Z2 ) ) )
=> ( ! [Z1: proces554692349s_term @ A @ Pvar] :
( ? [Z2: proces554692349s_term @ A @ Pvar] :
( A2
= ( proces89589571ess_CH @ A @ Pvar @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( proces1264666542s_term @ A @ Pvar @ Z1 ) ) )
=> ~ ! [Z1: proces554692349s_term @ A @ Pvar,Z2: proces554692349s_term @ A @ Pvar] :
( ( A2
= ( proces89589571ess_CH @ A @ Pvar @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( proces1264666542s_term @ A @ Pvar @ Z2 ) ) ) ) ) ) ) ) ).
% process_term.set_cases(1)
thf(fact_90_process__term_Opred__set,axiom,
! [B: $tType,A: $tType] :
( ( proces2118920028s_term @ A @ B )
= ( ^ [P12: A > $o,P22: B > $o,X3: proces554692349s_term @ A @ B] :
( ! [Y6: A] :
( ( member @ A @ Y6 @ ( proces1264666542s_term @ A @ B @ X3 ) )
=> ( P12 @ Y6 ) )
& ! [Y6: B] :
( ( member @ B @ Y6 @ ( proces1493547885s_term @ A @ B @ X3 ) )
=> ( P22 @ Y6 ) ) ) ) ) ).
% process_term.pred_set
thf(fact_91_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_92_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_93_process_Opred__set,axiom,
! [A: $tType] :
( ( proces1205983068rocess @ A )
= ( ^ [P4: A > $o,X3: proces634752977rocess @ A] :
! [Y6: A] :
( ( member @ A @ Y6 @ ( proces1148692671rocess @ A @ X3 ) )
=> ( P4 @ Y6 ) ) ) ) ).
% process.pred_set
thf(fact_94_process_Orel__cong,axiom,
! [A: $tType,B: $tType,X6: proces634752977rocess @ A,Ya2: proces634752977rocess @ A,Y3: proces634752977rocess @ B,Xa2: proces634752977rocess @ B,R3: A > B > $o,Ra: A > B > $o] :
( ( X6 = Ya2 )
=> ( ( Y3 = Xa2 )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( proces1148692671rocess @ A @ Ya2 ) )
=> ( ( member @ B @ Yb @ ( proces1148692671rocess @ B @ Xa2 ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( proces749077512rocess @ A @ B @ R3 @ X6 @ Y3 )
= ( proces749077512rocess @ A @ B @ Ra @ Ya2 @ Xa2 ) ) ) ) ) ).
% process.rel_cong
thf(fact_95_process_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X6: proces634752977rocess @ A,Y3: proces634752977rocess @ B,Ra: A > B > $o] :
( ( proces749077512rocess @ A @ B @ R3 @ X6 @ Y3 )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( proces1148692671rocess @ A @ X6 ) )
=> ( ( member @ B @ Yb @ ( proces1148692671rocess @ B @ Y3 ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( proces749077512rocess @ A @ B @ Ra @ X6 @ Y3 ) ) ) ).
% process.rel_mono_strong
thf(fact_96_process_Orel__refl__strong,axiom,
! [A: $tType,X6: proces634752977rocess @ A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( proces1148692671rocess @ A @ X6 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( proces749077512rocess @ A @ A @ Ra @ X6 @ X6 ) ) ).
% process.rel_refl_strong
thf(fact_97_process_Opred__cong,axiom,
! [A: $tType,X6: proces634752977rocess @ A,Ya2: proces634752977rocess @ A,P: A > $o,Pa: A > $o] :
( ( X6 = Ya2 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( proces1148692671rocess @ A @ Ya2 ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( proces1205983068rocess @ A @ P @ X6 )
= ( proces1205983068rocess @ A @ Pa @ Ya2 ) ) ) ) ).
% process.pred_cong
thf(fact_98_process_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X6: proces634752977rocess @ A,Pa: A > $o] :
( ( proces1205983068rocess @ A @ P @ X6 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( proces1148692671rocess @ A @ X6 ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( proces1205983068rocess @ A @ Pa @ X6 ) ) ) ).
% process.pred_mono_strong
thf(fact_99_process__term_Osimps_I46_J,axiom,
! [Pvar: $tType,A: $tType,X2: proces634752977rocess @ A] :
( ( proces1264666542s_term @ A @ Pvar @ ( proces1062592052s_PROC @ A @ Pvar @ X2 ) )
= ( proces1148692671rocess @ A @ X2 ) ) ).
% process_term.simps(46)
thf(fact_100_process__term_Oset__intros_I1_J,axiom,
! [Pvar: $tType,A: $tType,X6: A,Aa2: proces634752977rocess @ A] :
( ( member @ A @ X6 @ ( proces1148692671rocess @ A @ Aa2 ) )
=> ( member @ A @ X6 @ ( proces1264666542s_term @ A @ Pvar @ ( proces1062592052s_PROC @ A @ Pvar @ Aa2 ) ) ) ) ).
% process_term.set_intros(1)
thf(fact_101_surj__pair,axiom,
! [A: $tType,B: $tType,P5: product_prod @ A @ B] :
? [X4: A,Y5: B] :
( P5
= ( product_Pair @ A @ B @ X4 @ Y5 ) ) ).
% surj_pair
thf(fact_102_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P5: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_103_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_104_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_105_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_106_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E3: E2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C2 @ ( product_Pair @ D @ E2 @ D2 @ E3 ) ) ) ) ) ).
% prod_cases5
thf(fact_107_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F5: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E3: E2,F6: F5] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E2 @ F5 ) @ D2 @ ( product_Pair @ E2 @ F5 @ E3 @ F6 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_108_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E2: $tType,F5: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E3: E2,F6: F5,G4: G3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E2 @ ( product_prod @ F5 @ G3 ) @ E3 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_109_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X6 ) ) ).
% prod_induct3
thf(fact_110_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X6 ) ) ).
% prod_induct4
thf(fact_111_prod__induct5,axiom,
! [E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E3: E2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E2 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E2 ) @ C2 @ ( product_Pair @ D @ E2 @ D2 @ E3 ) ) ) ) )
=> ( P @ X6 ) ) ).
% prod_induct5
thf(fact_112_prod__induct6,axiom,
! [F5: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E3: E2,F6: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ F5 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E2 @ F5 ) @ D2 @ ( product_Pair @ E2 @ F5 @ E3 @ F6 ) ) ) ) ) )
=> ( P @ X6 ) ) ).
% prod_induct6
thf(fact_113_prod__induct7,axiom,
! [G3: $tType,F5: $tType,E2: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) ) ) ) > $o,X6: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E3: E2,F6: F5,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E2 @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E2 @ ( product_prod @ F5 @ G3 ) @ E3 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
=> ( P @ X6 ) ) ).
% prod_induct7
thf(fact_114_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y3
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_115_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_116_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_117_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P4: A > $o] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( P4 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_118_ball__reg,axiom,
! [A: $tType,R3: set @ A,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ R3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ R3 )
=> ( P @ X4 ) )
=> ! [X7: A] :
( ( member @ A @ X7 @ R3 )
=> ( Q @ X7 ) ) ) ) ).
% ball_reg
thf(fact_119_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F7: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F7 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_120_curryI,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_121_curryD,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_122_curryE,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_123_process__term_Octr__transfer_I2_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o] : ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( proces749077512rocess @ A @ B @ R1 ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( proces1062592052s_PROC @ A @ Pvar ) @ ( proces1062592052s_PROC @ B @ C ) ) ).
% process_term.ctr_transfer(2)
thf(fact_124_process__term_Osimps_I47_J,axiom,
! [Pvar: $tType,A: $tType,X31: A,X32: proces554692349s_term @ A @ Pvar] :
( ( proces1264666542s_term @ A @ Pvar @ ( proces1454156180ss_ACT @ A @ Pvar @ X31 @ X32 ) )
= ( insert @ A @ X31 @ ( proces1264666542s_term @ A @ Pvar @ X32 ) ) ) ).
% process_term.simps(47)
thf(fact_125_process__term_Osimps_I45_J,axiom,
! [Pvar: $tType,A: $tType,X1: Pvar] :
( ( proces1264666542s_term @ A @ Pvar @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% process_term.simps(45)
thf(fact_126_process__term_Osimps_I50_J,axiom,
! [A: $tType,Pvar: $tType,X2: proces634752977rocess @ A] :
( ( proces1493547885s_term @ A @ Pvar @ ( proces1062592052s_PROC @ A @ Pvar @ X2 ) )
= ( bot_bot @ ( set @ Pvar ) ) ) ).
% process_term.simps(50)
thf(fact_127_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_128_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_129_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_130_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_131_insert__absorb2,axiom,
! [A: $tType,X6: A,A3: set @ A] :
( ( insert @ A @ X6 @ ( insert @ A @ X6 @ A3 ) )
= ( insert @ A @ X6 @ A3 ) ) ).
% insert_absorb2
thf(fact_132_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_133_insertCI,axiom,
! [A: $tType,A2: A,B6: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B6 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% insertCI
thf(fact_134_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_135_ball__empty,axiom,
! [A: $tType,P: A > $o,X7: A] :
( ( member @ A @ X7 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X7 ) ) ).
% ball_empty
thf(fact_136_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B7: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B7 ) )
& ~ ( member @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_137_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_138_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_139_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B2 = D3 ) )
| ( ( A2 = D3 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_140_insert__commute,axiom,
! [A: $tType,X6: A,Y3: A,A3: set @ A] :
( ( insert @ A @ X6 @ ( insert @ A @ Y3 @ A3 ) )
= ( insert @ A @ Y3 @ ( insert @ A @ X6 @ A3 ) ) ) ).
% insert_commute
thf(fact_141_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_142_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A,B6: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B2 @ B6 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ B6 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B6 ) )
& ( ( A2 != B2 )
=> ? [C4: set @ A] :
( ( A3
= ( insert @ A @ B2 @ C4 ) )
& ~ ( member @ A @ B2 @ C4 )
& ( B6
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_143_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_144_insert__ident,axiom,
! [A: $tType,X6: A,A3: set @ A,B6: set @ A] :
( ~ ( member @ A @ X6 @ A3 )
=> ( ~ ( member @ A @ X6 @ B6 )
=> ( ( ( insert @ A @ X6 @ A3 )
= ( insert @ A @ X6 @ B6 ) )
= ( A3 = B6 ) ) ) ) ).
% insert_ident
thf(fact_145_let__rsp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o] :
( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ R1 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 ) @ R2 )
@ ^ [S3: A,F7: A > C] : ( F7 @ S3 )
@ ^ [S3: B,F7: B > D] : ( F7 @ S3 ) ) ).
% let_rsp
thf(fact_146_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_147_Set_Oset__insert,axiom,
! [A: $tType,X6: A,A3: set @ A] :
( ( member @ A @ X6 @ A3 )
=> ~ ! [B7: set @ A] :
( ( A3
= ( insert @ A @ X6 @ B7 ) )
=> ( member @ A @ X6 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_148_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_149_insertI2,axiom,
! [A: $tType,A2: A,B6: set @ A,B2: A] :
( ( member @ A @ A2 @ B6 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B6 ) ) ) ).
% insertI2
thf(fact_150_insertI1,axiom,
! [A: $tType,A2: A,B6: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B6 ) ) ).
% insertI1
thf(fact_151_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y5: A] :
~ ( member @ A @ Y5 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_152_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_153_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_154_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_155_process__term_Osimps_I49_J,axiom,
! [A: $tType,Pvar: $tType,X1: Pvar] :
( ( proces1493547885s_term @ A @ Pvar @ ( proces1627516585ss_VAR @ Pvar @ A @ X1 ) )
= ( insert @ Pvar @ X1 @ ( bot_bot @ ( set @ Pvar ) ) ) ) ).
% process_term.simps(49)
thf(fact_156_process__term_Orec__transfer,axiom,
! [Pvar: $tType,D: $tType,A: $tType,B: $tType,E2: $tType,C: $tType,R2: Pvar > C > $o,S4: D > E2 > $o,R1: A > B > $o] : ( bNF_rel_fun @ ( Pvar > D ) @ ( C > E2 ) @ ( ( ( proces634752977rocess @ A ) > D ) > ( A > ( proces554692349s_term @ A @ Pvar ) > D > D ) > ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D ) > ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( ( proces634752977rocess @ B ) > E2 ) > ( B > ( proces554692349s_term @ B @ C ) > E2 > E2 ) > ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) > E2 > E2 > E2 ) > ( proces554692349s_term @ B @ C ) > E2 ) @ ( bNF_rel_fun @ Pvar @ C @ D @ E2 @ R2 @ S4 ) @ ( bNF_rel_fun @ ( ( proces634752977rocess @ A ) > D ) @ ( ( proces634752977rocess @ B ) > E2 ) @ ( ( A > ( proces554692349s_term @ A @ Pvar ) > D > D ) > ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D ) > ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( B > ( proces554692349s_term @ B @ C ) > E2 > E2 ) > ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) > E2 > E2 > E2 ) > ( proces554692349s_term @ B @ C ) > E2 ) @ ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ D @ E2 @ ( proces749077512rocess @ A @ B @ R1 ) @ S4 ) @ ( bNF_rel_fun @ ( A > ( proces554692349s_term @ A @ Pvar ) > D > D ) @ ( B > ( proces554692349s_term @ B @ C ) > E2 > E2 ) @ ( ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D ) > ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) > E2 > E2 > E2 ) > ( proces554692349s_term @ B @ C ) > E2 ) @ ( bNF_rel_fun @ A @ B @ ( ( proces554692349s_term @ A @ Pvar ) > D > D ) @ ( ( proces554692349s_term @ B @ C ) > E2 > E2 ) @ R1 @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( D > D ) @ ( E2 > E2 ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ D @ E2 @ D @ E2 @ S4 @ S4 ) ) ) @ ( bNF_rel_fun @ ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D > D > D ) @ ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) > E2 > E2 > E2 ) @ ( ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( proces554692349s_term @ B @ C ) > E2 ) @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( ( proces554692349s_term @ A @ Pvar ) > D > D > D ) @ ( ( proces554692349s_term @ B @ C ) > E2 > E2 > E2 ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( D > D > D ) @ ( E2 > E2 > E2 ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ D @ E2 @ ( D > D ) @ ( E2 > E2 ) @ S4 @ ( bNF_rel_fun @ D @ E2 @ D @ E2 @ S4 @ S4 ) ) ) ) @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ D @ E2 @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ S4 ) ) ) ) @ ( proces2117273769s_term @ Pvar @ D @ A ) @ ( proces2117273769s_term @ C @ E2 @ B ) ) ).
% process_term.rec_transfer
thf(fact_157_process__term_Ocase__transfer,axiom,
! [Pvar: $tType,D: $tType,A: $tType,B: $tType,E2: $tType,C: $tType,R2: Pvar > C > $o,S4: D > E2 > $o,R1: A > B > $o] : ( bNF_rel_fun @ ( Pvar > D ) @ ( C > E2 ) @ ( ( ( proces634752977rocess @ A ) > D ) > ( A > ( proces554692349s_term @ A @ Pvar ) > D ) > ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D ) > ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( ( proces634752977rocess @ B ) > E2 ) > ( B > ( proces554692349s_term @ B @ C ) > E2 ) > ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) > E2 ) > ( proces554692349s_term @ B @ C ) > E2 ) @ ( bNF_rel_fun @ Pvar @ C @ D @ E2 @ R2 @ S4 ) @ ( bNF_rel_fun @ ( ( proces634752977rocess @ A ) > D ) @ ( ( proces634752977rocess @ B ) > E2 ) @ ( ( A > ( proces554692349s_term @ A @ Pvar ) > D ) > ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D ) > ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( B > ( proces554692349s_term @ B @ C ) > E2 ) > ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) > E2 ) > ( proces554692349s_term @ B @ C ) > E2 ) @ ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ D @ E2 @ ( proces749077512rocess @ A @ B @ R1 ) @ S4 ) @ ( bNF_rel_fun @ ( A > ( proces554692349s_term @ A @ Pvar ) > D ) @ ( B > ( proces554692349s_term @ B @ C ) > E2 ) @ ( ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D ) > ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) > E2 ) > ( proces554692349s_term @ B @ C ) > E2 ) @ ( bNF_rel_fun @ A @ B @ ( ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( proces554692349s_term @ B @ C ) > E2 ) @ R1 @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ D @ E2 @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ S4 ) ) @ ( bNF_rel_fun @ ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) > E2 ) @ ( ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( proces554692349s_term @ B @ C ) > E2 ) @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( ( proces554692349s_term @ A @ Pvar ) > D ) @ ( ( proces554692349s_term @ B @ C ) > E2 ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ D @ E2 @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ S4 ) ) @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ D @ E2 @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ S4 ) ) ) ) @ ( proces460752237s_term @ Pvar @ D @ A ) @ ( proces460752237s_term @ C @ E2 @ B ) ) ).
% process_term.case_transfer
thf(fact_158_process__term_Omap__transfer,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,L: $tType,J: $tType,K: $tType,I: $tType,R1b: A > I > $o,S1d: C > K > $o,R2b: B > J > $o,S2d: D > L > $o] : ( bNF_rel_fun @ ( A > C ) @ ( I > K ) @ ( ( B > D ) > ( proces554692349s_term @ A @ B ) > ( proces554692349s_term @ C @ D ) ) @ ( ( J > L ) > ( proces554692349s_term @ I @ J ) > ( proces554692349s_term @ K @ L ) ) @ ( bNF_rel_fun @ A @ I @ C @ K @ R1b @ S1d ) @ ( bNF_rel_fun @ ( B > D ) @ ( J > L ) @ ( ( proces554692349s_term @ A @ B ) > ( proces554692349s_term @ C @ D ) ) @ ( ( proces554692349s_term @ I @ J ) > ( proces554692349s_term @ K @ L ) ) @ ( bNF_rel_fun @ B @ J @ D @ L @ R2b @ S2d ) @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ B ) @ ( proces554692349s_term @ I @ J ) @ ( proces554692349s_term @ C @ D ) @ ( proces554692349s_term @ K @ L ) @ ( proces2029722208s_term @ A @ I @ B @ J @ R1b @ R2b ) @ ( proces2029722208s_term @ C @ K @ D @ L @ S1d @ S2d ) ) ) @ ( proces224683037s_term @ A @ C @ B @ D ) @ ( proces224683037s_term @ I @ K @ J @ L ) ) ).
% process_term.map_transfer
thf(fact_159_process__term_Octr__transfer_I1_J,axiom,
! [Pvar: $tType,A: $tType,B: $tType,C: $tType,R2: Pvar > C > $o,R1: A > B > $o] : ( bNF_rel_fun @ Pvar @ C @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ R2 @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( proces1627516585ss_VAR @ Pvar @ A ) @ ( proces1627516585ss_VAR @ C @ B ) ) ).
% process_term.ctr_transfer(1)
thf(fact_160_the__elem__eq,axiom,
! [A: $tType,X6: A] :
( ( the_elem @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
= X6 ) ).
% the_elem_eq
thf(fact_161_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B6: C > D > $o,F: A > C,G: B > D] :
( ! [X4: A,Y5: B] :
( ( A3 @ X4 @ Y5 )
=> ( B6 @ ( F @ X4 ) @ ( G @ Y5 ) ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B6 @ F @ G ) ) ).
% rel_funI
thf(fact_162_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X3: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_163_is__singletonI,axiom,
! [A: $tType,X6: A] : ( is_singleton @ A @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_164_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_165_process_Opred__transfer,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( proces634752977rocess @ A ) > $o ) @ ( ( proces634752977rocess @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R3
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ $o @ $o @ ( proces749077512rocess @ A @ B @ R3 )
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( proces1205983068rocess @ A )
@ ( proces1205983068rocess @ B ) ) ).
% process.pred_transfer
thf(fact_166_process_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ B ) > $o ) @ ( ( proces634752977rocess @ C ) > ( proces634752977rocess @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ C ) @ ( ( proces634752977rocess @ B ) > $o ) @ ( ( proces634752977rocess @ D ) > $o ) @ ( proces749077512rocess @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( proces634752977rocess @ B ) @ ( proces634752977rocess @ D ) @ $o @ $o @ ( proces749077512rocess @ B @ D @ Sc )
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( proces749077512rocess @ A @ B )
@ ( proces749077512rocess @ C @ D ) ) ).
% process.rel_transfer
thf(fact_167_process__term_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( C > $o ) @ ( ( B > $o ) > ( proces554692349s_term @ A @ B ) > $o ) @ ( ( D > $o ) > ( proces554692349s_term @ C @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ $o @ $o @ R1
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( B > $o ) @ ( D > $o ) @ ( ( proces554692349s_term @ A @ B ) > $o ) @ ( ( proces554692349s_term @ C @ D ) > $o )
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ R2
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( proces554692349s_term @ A @ B ) @ ( proces554692349s_term @ C @ D ) @ $o @ $o @ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 )
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( proces2118920028s_term @ A @ B )
@ ( proces2118920028s_term @ C @ D ) ) ).
% process_term.pred_transfer
thf(fact_168_process__term_Orel__transfer,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,H: $tType,F5: $tType,G3: $tType,E2: $tType,S1a: A > E2 > $o,S1c: C > G3 > $o,S2a: B > F5 > $o,S2c: D > H > $o] :
( bNF_rel_fun @ ( A > C > $o ) @ ( E2 > G3 > $o ) @ ( ( B > D > $o ) > ( proces554692349s_term @ A @ B ) > ( proces554692349s_term @ C @ D ) > $o ) @ ( ( F5 > H > $o ) > ( proces554692349s_term @ E2 @ F5 ) > ( proces554692349s_term @ G3 @ H ) > $o )
@ ( bNF_rel_fun @ A @ E2 @ ( C > $o ) @ ( G3 > $o ) @ S1a
@ ( bNF_rel_fun @ C @ G3 @ $o @ $o @ S1c
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( B > D > $o ) @ ( F5 > H > $o ) @ ( ( proces554692349s_term @ A @ B ) > ( proces554692349s_term @ C @ D ) > $o ) @ ( ( proces554692349s_term @ E2 @ F5 ) > ( proces554692349s_term @ G3 @ H ) > $o )
@ ( bNF_rel_fun @ B @ F5 @ ( D > $o ) @ ( H > $o ) @ S2a
@ ( bNF_rel_fun @ D @ H @ $o @ $o @ S2c
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( proces554692349s_term @ A @ B ) @ ( proces554692349s_term @ E2 @ F5 ) @ ( ( proces554692349s_term @ C @ D ) > $o ) @ ( ( proces554692349s_term @ G3 @ H ) > $o ) @ ( proces2029722208s_term @ A @ E2 @ B @ F5 @ S1a @ S2a )
@ ( bNF_rel_fun @ ( proces554692349s_term @ C @ D ) @ ( proces554692349s_term @ G3 @ H ) @ $o @ $o @ ( proces2029722208s_term @ C @ G3 @ D @ H @ S1c @ S2c )
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) ) )
@ ( proces2029722208s_term @ A @ C @ B @ D )
@ ( proces2029722208s_term @ E2 @ G3 @ F5 @ H ) ) ).
% process_term.rel_transfer
thf(fact_169_process__term_Octr__transfer_I4_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o] : ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) ) @ ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) ) @ ( proces89589571ess_CH @ A @ Pvar ) @ ( proces89589571ess_CH @ B @ C ) ) ).
% process_term.ctr_transfer(4)
thf(fact_170_process__term_Octr__transfer_I3_J,axiom,
! [A: $tType,Pvar: $tType,C: $tType,B: $tType,R1: A > B > $o,R2: Pvar > C > $o] : ( bNF_rel_fun @ A @ B @ ( ( proces554692349s_term @ A @ Pvar ) > ( proces554692349s_term @ A @ Pvar ) ) @ ( ( proces554692349s_term @ B @ C ) > ( proces554692349s_term @ B @ C ) ) @ R1 @ ( bNF_rel_fun @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( proces554692349s_term @ A @ Pvar ) @ ( proces554692349s_term @ B @ C ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) @ ( proces2029722208s_term @ A @ B @ Pvar @ C @ R1 @ R2 ) ) @ ( proces1454156180ss_ACT @ A @ Pvar ) @ ( proces1454156180ss_ACT @ B @ C ) ) ).
% process_term.ctr_transfer(3)
thf(fact_171_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
( A6
= ( insert @ A @ ( the_elem @ A @ A6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_172_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y5: A] :
( ( member @ A @ X4 @ A3 )
=> ( ( member @ A @ Y5 @ A3 )
=> ( X4 = Y5 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_173_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_174_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y7: A > B > $o,X8: A > B > $o,A3: C > D > $o,B6: C > D > $o,F: A > C,G: B > D] :
( ! [X4: A,Y5: B] :
( ( Y7 @ X4 @ Y5 )
=> ( X8 @ X4 @ Y5 ) )
=> ( ! [X4: C,Y5: D] :
( ( A3 @ X4 @ Y5 )
=> ( B6 @ X4 @ Y5 ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X8 @ A3 @ F @ G )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y7 @ B6 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_175_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X8: A > B > $o,A3: C > D > $o,F: A > C,G: B > D,Y7: A > B > $o,B6: C > D > $o] :
( ( bNF_rel_fun @ A @ B @ C @ D @ X8 @ A3 @ F @ G )
=> ( ! [X4: A,Y5: B] :
( ( Y7 @ X4 @ Y5 )
=> ( X8 @ X4 @ Y5 ) )
=> ( ! [X4: C,Y5: D] :
( ( A3 @ X4 @ Y5 )
=> ( B6 @ X4 @ Y5 ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y7 @ B6 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_176_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B6: C > D > $o,F: A > C,G: B > D,X6: A,Y3: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B6 @ F @ G )
=> ( ( A3 @ X6 @ Y3 )
=> ( B6 @ ( F @ X6 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funD
thf(fact_177_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_178_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X3: A] :
( A6
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_179_Let__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B6: C > D > $o] :
( bNF_rel_fun @ A @ B @ ( ( A > C ) > C ) @ ( ( B > D ) > D ) @ A3 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ C @ D @ ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B6 ) @ B6 )
@ ^ [S3: A,F7: A > C] : ( F7 @ S3 )
@ ^ [S3: B,F7: B > D] : ( F7 @ S3 ) ) ).
% Let_transfer
thf(fact_180_rel__fun__def__butlast,axiom,
! [B: $tType,D: $tType,C: $tType,E2: $tType,F5: $tType,A: $tType,R3: A > B > $o,S4: C > E2 > $o,T2: D > F5 > $o,F: A > C > D,G: B > E2 > F5] :
( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E2 > F5 ) @ R3 @ ( bNF_rel_fun @ C @ E2 @ D @ F5 @ S4 @ T2 ) @ F @ G )
= ( ! [X3: A,Y6: B] :
( ( R3 @ X3 @ Y6 )
=> ( bNF_rel_fun @ C @ E2 @ D @ F5 @ S4 @ T2 @ ( F @ X3 ) @ ( G @ Y6 ) ) ) ) ) ).
% rel_fun_def_butlast
thf(fact_181_If__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A3 @ ( bNF_rel_fun @ A @ B @ A @ B @ A3 @ A3 ) )
@ ( if @ A )
@ ( if @ B ) ) ).
% If_transfer
thf(fact_182_rel__funE,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B6: C > D > $o,F: A > C,G: B > D,X6: A,Y3: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B6 @ F @ G )
=> ( ( A3 @ X6 @ Y3 )
=> ( B6 @ ( F @ X6 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funE
thf(fact_183_rel__funD2,axiom,
! [B: $tType,C: $tType,A: $tType,A3: A > A > $o,B6: B > C > $o,F: A > B,G: A > C,X6: A] :
( ( bNF_rel_fun @ A @ A @ B @ C @ A3 @ B6 @ F @ G )
=> ( ( A3 @ X6 @ X6 )
=> ( B6 @ ( F @ X6 ) @ ( G @ X6 ) ) ) ) ).
% rel_funD2
thf(fact_184_fun_Orel__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,E2: $tType,D: $tType,Sa: A > C > $o,Sc: B > E2 > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > E2 > $o ) @ ( ( D > A ) > ( D > B ) > $o ) @ ( ( D > C ) > ( D > E2 ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( E2 > $o ) @ Sa
@ ( bNF_rel_fun @ B @ E2 @ $o @ $o @ Sc
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > C ) @ ( ( D > B ) > $o ) @ ( ( D > E2 ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y4: D,Z: D] : Y4 = Z
@ Sa )
@ ( bNF_rel_fun @ ( D > B ) @ ( D > E2 ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ B @ E2
@ ^ [Y4: D,Z: D] : Y4 = Z
@ Sc )
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z: D] : Y4 = Z )
@ ( bNF_rel_fun @ D @ D @ C @ E2
@ ^ [Y4: D,Z: D] : Y4 = Z ) ) ).
% fun.rel_transfer
thf(fact_185_process_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,S4: C > D > $o] : ( bNF_rel_fun @ ( A > ( proces634752977rocess @ A ) > C ) @ ( B > ( proces634752977rocess @ B ) > D ) @ ( ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > C ) > ( proces634752977rocess @ A ) > C ) @ ( ( ( proces634752977rocess @ B ) > ( proces634752977rocess @ B ) > D ) > ( proces634752977rocess @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( proces634752977rocess @ A ) > C ) @ ( ( proces634752977rocess @ B ) > D ) @ R3 @ ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ C @ D @ ( proces749077512rocess @ A @ B @ R3 ) @ S4 ) ) @ ( bNF_rel_fun @ ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > C ) @ ( ( proces634752977rocess @ B ) > ( proces634752977rocess @ B ) > D ) @ ( ( proces634752977rocess @ A ) > C ) @ ( ( proces634752977rocess @ B ) > D ) @ ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ ( ( proces634752977rocess @ A ) > C ) @ ( ( proces634752977rocess @ B ) > D ) @ ( proces749077512rocess @ A @ B @ R3 ) @ ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ C @ D @ ( proces749077512rocess @ A @ B @ R3 ) @ S4 ) ) @ ( bNF_rel_fun @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ C @ D @ ( proces749077512rocess @ A @ B @ R3 ) @ S4 ) ) @ ( proces1406508781rocess @ A @ C ) @ ( proces1406508781rocess @ B @ D ) ) ).
% process.case_transfer
thf(fact_186_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_187_fun_Orel__refl,axiom,
! [B: $tType,D: $tType,Ra: B > B > $o,X6: D > B] :
( ! [X4: B] : ( Ra @ X4 @ X4 )
=> ( bNF_rel_fun @ D @ D @ B @ B
@ ^ [Y4: D,Z: D] : Y4 = Z
@ Ra
@ X6
@ X6 ) ) ).
% fun.rel_refl
thf(fact_188_fun_Orel__eq,axiom,
! [A: $tType,D: $tType] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y4: D,Z: D] : Y4 = Z
@ ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [Y4: D > A,Z: D > A] : Y4 = Z ) ) ).
% fun.rel_eq
thf(fact_189_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_190_map__fun__parametric,axiom,
! [A: $tType,B: $tType,E2: $tType,F5: $tType,H: $tType,G3: $tType,D: $tType,C: $tType,A3: A > C > $o,B6: B > D > $o,C5: E2 > G3 > $o,D4: F5 > H > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E2 > F5 ) > ( B > E2 ) > A > F5 ) @ ( ( G3 > H ) > ( D > G3 ) > C > H ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A3 @ B6 ) @ ( bNF_rel_fun @ ( E2 > F5 ) @ ( G3 > H ) @ ( ( B > E2 ) > A > F5 ) @ ( ( D > G3 ) > C > H ) @ ( bNF_rel_fun @ E2 @ G3 @ F5 @ H @ C5 @ D4 ) @ ( bNF_rel_fun @ ( B > E2 ) @ ( D > G3 ) @ ( A > F5 ) @ ( C > H ) @ ( bNF_rel_fun @ B @ D @ E2 @ G3 @ B6 @ C5 ) @ ( bNF_rel_fun @ A @ C @ F5 @ H @ A3 @ D4 ) ) ) @ ( map_fun @ A @ B @ E2 @ F5 ) @ ( map_fun @ C @ D @ G3 @ H ) ) ).
% map_fun_parametric
thf(fact_191_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X6: A,Y3: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X6 @ Y3 ) )
= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_192_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X6: A,Y3: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X6 @ Y3 ) )
= ( insert @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_193_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_194_prod_Opred__set,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P12: A > $o,P22: B > $o,X3: product_prod @ A @ B] :
( ! [Y6: A] :
( ( member @ A @ Y6 @ ( basic_fsts @ A @ B @ X3 ) )
=> ( P12 @ Y6 ) )
& ! [Y6: B] :
( ( member @ B @ Y6 @ ( basic_snds @ A @ B @ X3 ) )
=> ( P22 @ Y6 ) ) ) ) ) ).
% prod.pred_set
thf(fact_195_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P1: A > $o,P2: B > $o,A2: A,B2: B] :
( ( basic_pred_prod @ A @ B @ P1 @ P2 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( ( P1 @ A2 )
& ( P2 @ B2 ) ) ) ).
% pred_prod_inject
thf(fact_196_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P1: A > $o,P2: B > $o,A2: product_prod @ A @ B] :
( ( basic_pred_prod @ A @ B @ P1 @ P2 @ A2 )
=> ~ ! [A4: A,B4: B] :
( ( A2
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( ( P1 @ A4 )
=> ~ ( P2 @ B4 ) ) ) ) ).
% pred_prod.cases
thf(fact_197_pred__prod_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P12: A > $o,P22: B > $o,A7: product_prod @ A @ B] :
? [B5: A,C6: B] :
( ( A7
= ( product_Pair @ A @ B @ B5 @ C6 ) )
& ( P12 @ B5 )
& ( P22 @ C6 ) ) ) ) ).
% pred_prod.simps
thf(fact_198_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P1: A > $o,A2: A,P2: B > $o,B2: B] :
( ( P1 @ A2 )
=> ( ( P2 @ B2 )
=> ( basic_pred_prod @ A @ B @ P1 @ P2 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ) ).
% pred_prod.intros
thf(fact_199_pred__prod_Oinducts,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,X6: product_prod @ A @ B,P: ( product_prod @ A @ B ) > $o] :
( ( basic_pred_prod @ A @ B @ P1 @ P2 @ X6 )
=> ( ! [A4: A,B4: B] :
( ( P1 @ A4 )
=> ( ( P2 @ B4 )
=> ( P @ ( product_Pair @ A @ B @ A4 @ B4 ) ) ) )
=> ( P @ X6 ) ) ) ).
% pred_prod.inducts
thf(fact_200_pred__prod__split,axiom,
! [B: $tType,A: $tType,P: $o > $o,Q: A > $o,R3: B > $o,Xy: product_prod @ A @ B] :
( ( P @ ( basic_pred_prod @ A @ B @ Q @ R3 @ Xy ) )
= ( ! [X3: A,Y6: B] :
( ( Xy
= ( product_Pair @ A @ B @ X3 @ Y6 ) )
=> ( P
@ ( ( Q @ X3 )
& ( R3 @ Y6 ) ) ) ) ) ) ).
% pred_prod_split
thf(fact_201_if__rsp,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( A > A > A )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ( bNF_rel_fun @ A @ A @ ( A > A ) @ ( A > A ) @ R3 @ ( bNF_rel_fun @ A @ A @ A @ A @ R3 @ R3 ) )
@ ( if @ A )
@ ( if @ A ) ) ) ).
% if_rsp
thf(fact_202_fun__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R2: C > C > $o,Abs2: C > D,Rep2: D > C] :
( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs2 @ Rep2 )
=> ( quotient3 @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 ) @ ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs2 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep2 ) ) ) ) ).
% fun_quotient3
thf(fact_203_apply__rspQ3_H_H,axiom,
! [C: $tType,A: $tType,B: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,S4: C > C > $o,F: A > C,X6: B] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R3 @ S4 @ F @ F )
=> ( S4 @ ( F @ ( Rep @ X6 ) ) @ ( F @ ( Rep @ X6 ) ) ) ) ) ).
% apply_rspQ3''
thf(fact_204_apply__rspQ3,axiom,
! [B: $tType,C: $tType,A: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,F: A > C,G: A > C,X6: A,Y3: A] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( bNF_rel_fun @ A @ A @ C @ C @ R1 @ R2 @ F @ G )
=> ( ( R1 @ X6 @ Y3 )
=> ( R2 @ ( F @ X6 ) @ ( G @ Y3 ) ) ) ) ) ).
% apply_rspQ3
thf(fact_205_quot__rel__rsp,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( bNF_rel_fun @ A @ A @ ( A > $o ) @ ( A > $o ) @ R3
@ ( bNF_rel_fun @ A @ A @ $o @ $o @ R3
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ R3
@ R3 ) ) ).
% quot_rel_rsp
thf(fact_206_let__prs,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > A > $o,Abs12: A > B,Rep12: B > A,R2: C > C > $o,Abs22: C > D,Rep22: D > C] :
( ( quotient3 @ A @ B @ R1 @ Abs12 @ Rep12 )
=> ( ( quotient3 @ C @ D @ R2 @ Abs22 @ Rep22 )
=> ( ( map_fun @ D @ C @ ( ( C > A ) > A ) @ ( ( D > B ) > B ) @ Rep22 @ ( map_fun @ ( D > B ) @ ( C > A ) @ A @ B @ ( map_fun @ C @ D @ B @ A @ Abs22 @ Rep12 ) @ Abs12 )
@ ^ [S3: C,F7: C > A] : ( F7 @ S3 ) )
= ( ^ [S3: D,F7: D > B] : ( F7 @ S3 ) ) ) ) ) ).
% let_prs
thf(fact_207_cond__prs,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,Absf: A > B,Repf: B > A,A2: $o,B2: B,C3: B] :
( ( quotient3 @ A @ B @ R3 @ Absf @ Repf )
=> ( ( A2
=> ( ( Absf @ ( if @ A @ A2 @ ( Repf @ B2 ) @ ( Repf @ C3 ) ) )
= B2 ) )
& ( ~ A2
=> ( ( Absf @ ( if @ A @ A2 @ ( Repf @ B2 ) @ ( Repf @ C3 ) ) )
= C3 ) ) ) ) ).
% cond_prs
thf(fact_208_Quotient3I,axiom,
! [B: $tType,A: $tType,Abs: B > A,Rep: A > B,R3: B > B > $o] :
( ! [A4: A] :
( ( Abs @ ( Rep @ A4 ) )
= A4 )
=> ( ! [A4: A] : ( R3 @ ( Rep @ A4 ) @ ( Rep @ A4 ) )
=> ( ! [R4: B,S5: B] :
( ( R3 @ R4 @ S5 )
= ( ( R3 @ R4 @ R4 )
& ( R3 @ S5 @ S5 )
& ( ( Abs @ R4 )
= ( Abs @ S5 ) ) ) )
=> ( quotient3 @ B @ A @ R3 @ Abs @ Rep ) ) ) ) ).
% Quotient3I
thf(fact_209_equals__rsp,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,Xa2: A,Xb: A,Ya2: A,Yb2: A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( R3 @ Xa2 @ Xb )
=> ( ( R3 @ Ya2 @ Yb2 )
=> ( ( R3 @ Xa2 @ Ya2 )
= ( R3 @ Xb @ Yb2 ) ) ) ) ) ).
% equals_rsp
thf(fact_210_rep__abs__rsp,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X2: A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( R3 @ X1 @ X2 )
=> ( R3 @ X1 @ ( Rep @ ( Abs @ X2 ) ) ) ) ) ).
% rep_abs_rsp
thf(fact_211_Quotient3__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient3 @ A @ B )
= ( ^ [R5: A > A > $o,Abs3: A > B,Rep3: B > A] :
( ! [A7: B] :
( ( Abs3 @ ( Rep3 @ A7 ) )
= A7 )
& ! [A7: B] : ( R5 @ ( Rep3 @ A7 ) @ ( Rep3 @ A7 ) )
& ! [R6: A,S3: A] :
( ( R5 @ R6 @ S3 )
= ( ( R5 @ R6 @ R6 )
& ( R5 @ S3 @ S3 )
& ( ( Abs3 @ R6 )
= ( Abs3 @ S3 ) ) ) ) ) ) ) ).
% Quotient3_def
thf(fact_212_Quotient3__rel,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,R: A,S: A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( ( R3 @ R @ R )
& ( R3 @ S @ S )
& ( ( Abs @ R )
= ( Abs @ S ) ) )
= ( R3 @ R @ S ) ) ) ).
% Quotient3_rel
thf(fact_213_Quotient3__refl1,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,R: A,S: A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( R3 @ R @ S )
=> ( R3 @ R @ R ) ) ) ).
% Quotient3_refl1
thf(fact_214_Quotient3__refl2,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,R: A,S: A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( R3 @ R @ S )
=> ( R3 @ S @ S ) ) ) ).
% Quotient3_refl2
thf(fact_215_rep__abs__rsp__left,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,X1: A,X2: A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( R3 @ X1 @ X2 )
=> ( R3 @ ( Rep @ ( Abs @ X1 ) ) @ X2 ) ) ) ).
% rep_abs_rsp_left
thf(fact_216_Quotient3__abs__rep,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,A2: B] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( Abs @ ( Rep @ A2 ) )
= A2 ) ) ).
% Quotient3_abs_rep
thf(fact_217_Quotient3__rel__abs,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,R: A,S: A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( R3 @ R @ S )
=> ( ( Abs @ R )
= ( Abs @ S ) ) ) ) ).
% Quotient3_rel_abs
thf(fact_218_Quotient3__rel__rep,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,A2: B,B2: B] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( R3 @ ( Rep @ A2 ) @ ( Rep @ B2 ) )
= ( A2 = B2 ) ) ) ).
% Quotient3_rel_rep
thf(fact_219_Quotient3__rep__abs,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,R: A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( R3 @ R @ R )
=> ( R3 @ ( Rep @ ( Abs @ R ) ) @ R ) ) ) ).
% Quotient3_rep_abs
thf(fact_220_Quotient3__rep__reflp,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,A2: B] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( R3 @ ( Rep @ A2 ) @ ( Rep @ A2 ) ) ) ).
% Quotient3_rep_reflp
thf(fact_221_apply__rsp_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > B > $o,R2: C > D > $o,F: A > C,G: B > D,X6: A,Y3: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ R1 @ R2 @ F @ G )
=> ( ( R1 @ X6 @ Y3 )
=> ( R2 @ ( F @ X6 ) @ ( G @ Y3 ) ) ) ) ).
% apply_rsp'
thf(fact_222_process_Ocorec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S4: C > D > $o,R3: A > B > $o] :
( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > A ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > B ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ S4
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ S4 @ R3 )
@ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ S4
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( C > ( proces634752977rocess @ A ) ) @ ( D > ( proces634752977rocess @ B ) ) @ ( ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ S4 @ ( proces749077512rocess @ A @ B @ R3 ) )
@ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S4 @ S4 )
@ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ S4
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( C > ( proces634752977rocess @ A ) ) @ ( D > ( proces634752977rocess @ B ) ) @ ( ( C > C ) > ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > D ) > ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ S4 @ ( proces749077512rocess @ A @ B @ R3 ) )
@ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( ( C > $o ) > ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > $o ) > ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S4 @ S4 )
@ ( bNF_rel_fun @ ( C > $o ) @ ( D > $o ) @ ( ( C > ( proces634752977rocess @ A ) ) > ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > ( proces634752977rocess @ B ) ) > ( D > D ) > D > ( proces634752977rocess @ B ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ S4
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( C > ( proces634752977rocess @ A ) ) @ ( D > ( proces634752977rocess @ B ) ) @ ( ( C > C ) > C > ( proces634752977rocess @ A ) ) @ ( ( D > D ) > D > ( proces634752977rocess @ B ) ) @ ( bNF_rel_fun @ C @ D @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ S4 @ ( proces749077512rocess @ A @ B @ R3 ) ) @ ( bNF_rel_fun @ ( C > C ) @ ( D > D ) @ ( C > ( proces634752977rocess @ A ) ) @ ( D > ( proces634752977rocess @ B ) ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S4 @ S4 ) @ ( bNF_rel_fun @ C @ D @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ S4 @ ( proces749077512rocess @ A @ B @ R3 ) ) ) ) ) ) ) ) ) ) ) )
@ ( proces894737309rocess @ C @ A )
@ ( proces894737309rocess @ D @ B ) ) ).
% process.corec_transfer
thf(fact_223_bex1__rel__rsp,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Absf: A > B,Repf: B > A] :
( ( quotient3 @ A @ B @ R3 @ Absf @ Repf )
=> ( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ A @ $o @ $o @ R3
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ( bex1_rel @ A @ R3 )
@ ( bex1_rel @ A @ R3 ) ) ) ).
% bex1_rel_rsp
thf(fact_224_bex1__rel__aux2,axiom,
! [A: $tType,R3: A > A > $o,X6: A > $o,Y3: A > $o] :
( ! [Xa: A,Ya: A] :
( ( R3 @ Xa @ Ya )
=> ( ( X6 @ Xa )
= ( Y3 @ Ya ) ) )
=> ( ( bex1_rel @ A @ R3 @ Y3 )
=> ( bex1_rel @ A @ R3 @ X6 ) ) ) ).
% bex1_rel_aux2
thf(fact_225_bex1__rel__aux,axiom,
! [A: $tType,R3: A > A > $o,X6: A > $o,Y3: A > $o] :
( ! [Xa: A,Ya: A] :
( ( R3 @ Xa @ Ya )
=> ( ( X6 @ Xa )
= ( Y3 @ Ya ) ) )
=> ( ( bex1_rel @ A @ R3 @ X6 )
=> ( bex1_rel @ A @ R3 @ Y3 ) ) ) ).
% bex1_rel_aux
thf(fact_226_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_227_fun__ord__parametric,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,F5: $tType,E2: $tType,C5: A > B > $o,A3: C > E2 > $o,B6: D > F5 > $o] :
( ( bi_total @ A @ B @ C5 )
=> ( bNF_rel_fun @ ( C > D > $o ) @ ( E2 > F5 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E2 ) > ( B > F5 ) > $o )
@ ( bNF_rel_fun @ C @ E2 @ ( D > $o ) @ ( F5 > $o ) @ A3
@ ( bNF_rel_fun @ D @ F5 @ $o @ $o @ B6
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > E2 ) @ ( ( A > D ) > $o ) @ ( ( B > F5 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E2 @ C5 @ A3 )
@ ( bNF_rel_fun @ ( A > D ) @ ( B > F5 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F5 @ C5 @ B6 )
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( partial_fun_ord @ C @ D @ A )
@ ( partial_fun_ord @ E2 @ F5 @ B ) ) ) ).
% fun_ord_parametric
thf(fact_228_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_229_pairwise__insert,axiom,
! [A: $tType,R: A > A > $o,X6: A,S: set @ A] :
( ( pairwise @ A @ R @ ( insert @ A @ X6 @ S ) )
= ( ! [Y6: A] :
( ( ( member @ A @ Y6 @ S )
& ( Y6 != X6 ) )
=> ( ( R @ X6 @ Y6 )
& ( R @ Y6 @ X6 ) ) )
& ( pairwise @ A @ R @ S ) ) ) ).
% pairwise_insert
thf(fact_230_process__term_Obi__total__rel,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > C > $o,R2: B > D > $o] :
( ( bi_total @ A @ C @ R1 )
=> ( ( bi_total @ B @ D @ R2 )
=> ( bi_total @ ( proces554692349s_term @ A @ B ) @ ( proces554692349s_term @ C @ D ) @ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 ) ) ) ) ).
% process_term.bi_total_rel
thf(fact_231_process_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( bi_total @ A @ B @ R3 )
=> ( bi_total @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ ( proces749077512rocess @ A @ B @ R3 ) ) ) ).
% process.bi_total_rel
thf(fact_232_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R5: A > A > $o,S6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ S6 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ S6 )
=> ( ( X3 != Y6 )
=> ( R5 @ X3 @ Y6 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_233_monotone__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B6: C > D > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A3
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
@ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B6
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B6
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B6 )
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( comple1396247847notone @ A @ C )
@ ( comple1396247847notone @ B @ D ) ) ) ).
% monotone_parametric
thf(fact_234_transfer__forall__transfer_I1_J,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ( transfer_forall @ A )
@ ( transfer_forall @ B ) ) ) ).
% transfer_forall_transfer(1)
thf(fact_235_transfer__forall__transfer_I4_J,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ rev_implies
@ ( transfer_forall @ A )
@ ( transfer_forall @ B ) ) ) ).
% transfer_forall_transfer(4)
thf(fact_236_transfer__forall__transfer_I5_J,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3 @ rev_implies ) @ rev_implies @ ( transfer_forall @ A ) @ ( transfer_forall @ B ) ) ) ).
% transfer_forall_transfer(5)
thf(fact_237_relcompp__transfer,axiom,
! [C: $tType,A: $tType,E2: $tType,F5: $tType,B: $tType,D: $tType,B6: A > B > $o,A3: C > D > $o,C5: E2 > F5 > $o] :
( ( bi_total @ A @ B @ B6 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E2 > $o ) > C > E2 > $o ) @ ( ( B > F5 > $o ) > D > F5 > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A3
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B6
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( A > E2 > $o ) @ ( B > F5 > $o ) @ ( C > E2 > $o ) @ ( D > F5 > $o )
@ ( bNF_rel_fun @ A @ B @ ( E2 > $o ) @ ( F5 > $o ) @ B6
@ ( bNF_rel_fun @ E2 @ F5 @ $o @ $o @ C5
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ C @ D @ ( E2 > $o ) @ ( F5 > $o ) @ A3
@ ( bNF_rel_fun @ E2 @ F5 @ $o @ $o @ C5
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) ) )
@ ( relcompp @ C @ A @ E2 )
@ ( relcompp @ D @ B @ F5 ) ) ) ).
% relcompp_transfer
thf(fact_238_Domainp__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B6: A > B > $o,A3: C > D > $o] :
( ( bi_total @ A @ B @ B6 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A3
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B6
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ A3
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( domainp @ C @ A )
@ ( domainp @ D @ B ) ) ) ).
% Domainp_transfer
thf(fact_239_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R3: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( bot_bot @ ( A > C > $o ) ) @ R3 )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot1
thf(fact_240_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R3: A > C > $o] :
( ( relcompp @ A @ C @ B @ R3 @ ( bot_bot @ ( C > B > $o ) ) )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot2
thf(fact_241_eq__comp__r,axiom,
! [A: $tType,R3: A > A > $o] :
( ( relcompp @ A @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z
@ ( relcompp @ A @ A @ A @ R3
@ ^ [Y4: A,Z: A] : Y4 = Z ) )
= R3 ) ).
% eq_comp_r
thf(fact_242_process__term_ODomainp__rel,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > C > $o,R2: B > D > $o] :
( ( domainp @ ( proces554692349s_term @ A @ B ) @ ( proces554692349s_term @ C @ D ) @ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 ) )
= ( proces2118920028s_term @ A @ B @ ( domainp @ A @ C @ R1 ) @ ( domainp @ B @ D @ R2 ) ) ) ).
% process_term.Domainp_rel
thf(fact_243_process_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( domainp @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ ( proces749077512rocess @ A @ B @ R3 ) )
= ( proces1205983068rocess @ A @ ( domainp @ A @ B @ R3 ) ) ) ).
% process.Domainp_rel
thf(fact_244_process_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: A > B > $o,S4: B > C > $o] :
( ( proces749077512rocess @ A @ C @ ( relcompp @ A @ B @ C @ R3 @ S4 ) )
= ( relcompp @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ ( proces634752977rocess @ C ) @ ( proces749077512rocess @ A @ B @ R3 ) @ ( proces749077512rocess @ B @ C @ S4 ) ) ) ).
% process.rel_compp
thf(fact_245_process__term_Orel__compp,axiom,
! [B: $tType,A: $tType,C: $tType,E2: $tType,F5: $tType,D: $tType,R1: A > C > $o,S1: C > E2 > $o,R2: B > D > $o,S22: D > F5 > $o] :
( ( proces2029722208s_term @ A @ E2 @ B @ F5 @ ( relcompp @ A @ C @ E2 @ R1 @ S1 ) @ ( relcompp @ B @ D @ F5 @ R2 @ S22 ) )
= ( relcompp @ ( proces554692349s_term @ A @ B ) @ ( proces554692349s_term @ C @ D ) @ ( proces554692349s_term @ E2 @ F5 ) @ ( proces2029722208s_term @ A @ C @ B @ D @ R1 @ R2 ) @ ( proces2029722208s_term @ C @ E2 @ D @ F5 @ S1 @ S22 ) ) ) ).
% process_term.rel_compp
thf(fact_246_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: B > A,R: C > A > $o,S: A > D > $o,A2: C,C3: D] :
( ! [Y5: A] :
? [X7: B] :
( Y5
= ( F @ X7 ) )
=> ( ( relcompp @ C @ A @ D @ R @ S @ A2 @ C3 )
=> ~ ! [B4: B] :
( ( R @ A2 @ ( F @ B4 ) )
=> ~ ( S @ ( F @ B4 ) @ C3 ) ) ) ) ).
% nchotomy_relcomppE
thf(fact_247_fun_Orel__compp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,R3: A > B > $o,S4: B > C > $o] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y4: D,Z: D] : Y4 = Z
@ ( relcompp @ A @ B @ C @ R3 @ S4 ) )
= ( relcompp @ ( D > A ) @ ( D > B ) @ ( D > C )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y4: D,Z: D] : Y4 = Z
@ R3 )
@ ( bNF_rel_fun @ D @ D @ B @ C
@ ^ [Y4: D,Z: D] : Y4 = Z
@ S4 ) ) ) ).
% fun.rel_compp
thf(fact_248_reflp__transfer_I5_J,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A3
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ rev_implies
@ ( reflp @ A )
@ ( reflp @ B ) ) ) ).
% reflp_transfer(5)
thf(fact_249_reflp__transfer_I4_J,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A3 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3 @ rev_implies ) ) @ rev_implies @ ( reflp @ A ) @ ( reflp @ B ) ) ) ).
% reflp_transfer(4)
thf(fact_250_fun_Orel__reflp,axiom,
! [D: $tType,A: $tType,R3: A > A > $o] :
( ( reflp @ A @ R3 )
=> ( reflp @ ( D > A )
@ ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y4: D,Z: D] : Y4 = Z
@ R3 ) ) ) ).
% fun.rel_reflp
thf(fact_251_bot2E,axiom,
! [A: $tType,B: $tType,X6: A,Y3: B] :
~ ( bot_bot @ ( A > B > $o ) @ X6 @ Y3 ) ).
% bot2E
thf(fact_252_process__term_Orel__reflp,axiom,
! [A: $tType,B: $tType,R1: A > A > $o,R2: B > B > $o] :
( ( reflp @ A @ R1 )
=> ( ( reflp @ B @ R2 )
=> ( reflp @ ( proces554692349s_term @ A @ B ) @ ( proces2029722208s_term @ A @ A @ B @ B @ R1 @ R2 ) ) ) ) ).
% process_term.rel_reflp
thf(fact_253_process_Orel__reflp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( reflp @ A @ R3 )
=> ( reflp @ ( proces634752977rocess @ A ) @ ( proces749077512rocess @ A @ A @ R3 ) ) ) ).
% process.rel_reflp
thf(fact_254_reflp__transfer_I1_J,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A3
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ( reflp @ A )
@ ( reflp @ B ) ) ) ).
% reflp_transfer(1)
%----Type constructors (3)
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_1,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Obot_2,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X6: A,Y3: A] :
( ( if @ A @ $false @ X6 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X6: A,Y3: A] :
( ( if @ A @ $true @ X6 @ Y3 )
= X6 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( proces687458811_isACT @ a @ b @ a @ c @ sys @ ( proces1627516585ss_VAR @ a @ c @ x ) )
= ( proces687458811_isACT @ a @ b @ a @ b @ sys @ ( sys @ x ) ) ) ).
%------------------------------------------------------------------------------