TPTP Problem File: PRO019^1.p
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%------------------------------------------------------------------------------
% File : PRO019^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Processes
% Problem : Process algebra 25
% 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__25.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 328 ( 152 unt; 44 typ; 0 def)
% Number of atoms : 698 ( 354 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 4525 ( 60 ~; 14 |; 63 &;4096 @)
% ( 0 <=>; 292 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 512 ( 512 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 43 usr; 3 con; 0-14 aty)
% Number of variables : 1359 ( 48 ^;1236 !; 24 ?;1359 :)
% ( 51 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:39:38.993
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Process__Mirabelle__xovoopgpgi_Oprocess,type,
proces634752977rocess: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (40)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OAction,type,
proces1525233512Action:
!>[A: $tType] : ( A > ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OChoice,type,
proces1915862579Choice:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
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_Och1Of,type,
proces979765041_ch1Of:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Och2Of,type,
proces988026546_ch2Of:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OcontOf,type,
proces1778668539contOf:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) ) ).
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_OisAction,type,
proces10484146Action:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_OisChoice,type,
proces401113213Choice:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > $o ) ).
thf(sy_c_Process__Mirabelle__xovoopgpgi_Oprocess_Omap__process,type,
proces370407749rocess:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( proces634752977rocess @ A ) > ( proces634752977rocess @ Aa ) ) ).
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_OprefOf,type,
proces745025900prefOf:
!>[A: $tType] : ( ( proces634752977rocess @ A ) > A ) ).
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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
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_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_p,type,
p: proces634752977rocess @ a ).
%----Relevant facts (256)
thf(fact_0_process_Odistinct__disc_I2_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( proces401113213Choice @ A @ Process )
=> ~ ( proces10484146Action @ A @ Process ) ) ).
% process.distinct_disc(2)
thf(fact_1_process_Odistinct__disc_I1_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( proces10484146Action @ A @ Process )
=> ~ ( proces401113213Choice @ A @ Process ) ) ).
% process.distinct_disc(1)
thf(fact_2_process_Oexhaust__disc,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ~ ( proces10484146Action @ A @ Process )
=> ( proces401113213Choice @ A @ Process ) ) ).
% process.exhaust_disc
thf(fact_3_process_Ocorec__disc__iff_I2_J,axiom,
! [A: $tType,C: $tType,P: C > $o,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C,A2: C] :
( ( proces401113213Choice @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( ~ ( P @ A2 ) ) ) ).
% process.corec_disc_iff(2)
thf(fact_4_process_Ocorec__disc__iff_I1_J,axiom,
! [A: $tType,C: $tType,P: C > $o,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C,A2: C] :
( ( proces10484146Action @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( P @ A2 ) ) ).
% process.corec_disc_iff(1)
thf(fact_5_process_Omap__disc__iff_I2_J,axiom,
! [B: $tType,A: $tType,F: A > B,A2: proces634752977rocess @ A] :
( ( proces401113213Choice @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces401113213Choice @ A @ A2 ) ) ).
% process.map_disc_iff(2)
thf(fact_6_process_Omap__disc__iff_I1_J,axiom,
! [B: $tType,A: $tType,F: A > B,A2: proces634752977rocess @ A] :
( ( proces10484146Action @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces10484146Action @ A @ A2 ) ) ).
% process.map_disc_iff(1)
thf(fact_7_process_Ocorec__disc_I2_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C] :
( ~ ( P @ A2 )
=> ( proces401113213Choice @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) ) ) ).
% process.corec_disc(2)
thf(fact_8_process_Ocorec__disc_I1_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C] :
( ( P @ A2 )
=> ( proces10484146Action @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) ) ) ).
% process.corec_disc(1)
thf(fact_9_process_Odisc_I4_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] : ( proces401113213Choice @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) ) ).
% process.disc(4)
thf(fact_10_process_Odisc_I3_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] :
~ ( proces401113213Choice @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) ) ).
% process.disc(3)
thf(fact_11_process_OdiscI_I2_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( Process
= ( proces1915862579Choice @ A @ X21 @ X22 ) )
=> ( proces401113213Choice @ A @ Process ) ) ).
% process.discI(2)
thf(fact_12_isChoice__def,axiom,
! [A: $tType] :
( ( proces401113213Choice @ A )
= ( ^ [Process2: proces634752977rocess @ A] :
? [X212: proces634752977rocess @ A,X222: proces634752977rocess @ A] :
( Process2
= ( proces1915862579Choice @ A @ X212 @ X222 ) ) ) ) ).
% isChoice_def
thf(fact_13_process_Odisc_I2_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
~ ( proces10484146Action @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) ) ).
% process.disc(2)
thf(fact_14_process_Odisc_I1_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] : ( proces10484146Action @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) ) ).
% process.disc(1)
thf(fact_15_process_Oinject_I2_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A,Y21: proces634752977rocess @ A,Y22: proces634752977rocess @ A] :
( ( ( proces1915862579Choice @ A @ X21 @ X22 )
= ( proces1915862579Choice @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% process.inject(2)
thf(fact_16_process_Oinject_I1_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A,Y11: A,Y12: proces634752977rocess @ A] :
( ( ( proces1525233512Action @ A @ X11 @ X12 )
= ( proces1525233512Action @ A @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% process.inject(1)
thf(fact_17_process_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C] :
( ~ ( P @ A2 )
=> ( ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 )
= ( proces1915862579Choice @ A @ ( if @ ( proces634752977rocess @ A ) @ ( Q21 @ A2 ) @ ( G211 @ A2 ) @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ ( G212 @ A2 ) ) ) @ ( if @ ( proces634752977rocess @ A ) @ ( Q22 @ A2 ) @ ( G221 @ A2 ) @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ ( G222 @ A2 ) ) ) ) ) ) ).
% process.corec(2)
thf(fact_18_process_Ocorec_I1_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C] :
( ( P @ A2 )
=> ( ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 )
= ( proces1525233512Action @ A @ ( G11 @ A2 ) @ ( if @ ( proces634752977rocess @ A ) @ ( Q12 @ A2 ) @ ( G121 @ A2 ) @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ ( G122 @ A2 ) ) ) ) ) ) ).
% process.corec(1)
thf(fact_19_process_Osimps_I16_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces370407749rocess @ A @ B @ F @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= ( proces1915862579Choice @ B @ ( proces370407749rocess @ A @ B @ F @ X21 ) @ ( proces370407749rocess @ A @ B @ F @ X22 ) ) ) ).
% process.simps(16)
thf(fact_20_process_Osimps_I15_J,axiom,
! [B: $tType,A: $tType,F: A > B,X11: A,X12: proces634752977rocess @ A] :
( ( proces370407749rocess @ A @ B @ F @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= ( proces1525233512Action @ B @ ( F @ X11 ) @ ( proces370407749rocess @ A @ B @ F @ X12 ) ) ) ).
% process.simps(15)
thf(fact_21_process_Odistinct_I1_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces1525233512Action @ A @ X11 @ X12 )
!= ( proces1915862579Choice @ A @ X21 @ X22 ) ) ).
% process.distinct(1)
thf(fact_22_process_Oexhaust,axiom,
! [A: $tType,Y: proces634752977rocess @ A] :
( ! [X112: A,X122: proces634752977rocess @ A] :
( Y
!= ( proces1525233512Action @ A @ X112 @ X122 ) )
=> ~ ! [X213: proces634752977rocess @ A,X223: proces634752977rocess @ A] :
( Y
!= ( proces1915862579Choice @ A @ X213 @ X223 ) ) ) ).
% process.exhaust
thf(fact_23_isAction__def,axiom,
! [A: $tType] :
( ( proces10484146Action @ A )
= ( ^ [Process2: proces634752977rocess @ A] :
? [X113: A,X123: proces634752977rocess @ A] :
( Process2
= ( proces1525233512Action @ A @ X113 @ X123 ) ) ) ) ).
% isAction_def
thf(fact_24_process_OdiscI_I1_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A,X11: A,X12: proces634752977rocess @ A] :
( ( Process
= ( proces1525233512Action @ A @ X11 @ X12 ) )
=> ( proces10484146Action @ A @ Process ) ) ).
% process.discI(1)
thf(fact_25_process_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: proces634752977rocess @ A,F: A > B] :
( ( proces10484146Action @ A @ A2 )
=> ( ( proces745025900prefOf @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( F @ ( proces745025900prefOf @ A @ A2 ) ) ) ) ).
% process.map_sel(1)
thf(fact_26_process_Omap__sel_I3_J,axiom,
! [B: $tType,A: $tType,A2: proces634752977rocess @ A,F: A > B] :
( ( proces401113213Choice @ A @ A2 )
=> ( ( proces979765041_ch1Of @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces370407749rocess @ A @ B @ F @ ( proces979765041_ch1Of @ A @ A2 ) ) ) ) ).
% process.map_sel(3)
thf(fact_27_process_Omap__sel_I4_J,axiom,
! [B: $tType,A: $tType,A2: proces634752977rocess @ A,F: A > B] :
( ( proces401113213Choice @ A @ A2 )
=> ( ( proces988026546_ch2Of @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces370407749rocess @ A @ B @ F @ ( proces988026546_ch2Of @ A @ A2 ) ) ) ) ).
% process.map_sel(4)
thf(fact_28_process_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: proces634752977rocess @ A,F: A > B] :
( ( proces10484146Action @ A @ A2 )
=> ( ( proces1778668539contOf @ B @ ( proces370407749rocess @ A @ B @ F @ A2 ) )
= ( proces370407749rocess @ A @ B @ F @ ( proces1778668539contOf @ A @ A2 ) ) ) ) ).
% process.map_sel(2)
thf(fact_29_process_Oset__cases,axiom,
! [A: $tType,E2: A,A2: proces634752977rocess @ A] :
( ( member @ A @ E2 @ ( proces1148692671rocess @ A @ A2 ) )
=> ( ! [Z2: proces634752977rocess @ A] :
( A2
!= ( proces1525233512Action @ A @ E2 @ Z2 ) )
=> ( ! [Z1: A,Z2: proces634752977rocess @ A] :
( ( A2
= ( proces1525233512Action @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E2 @ ( proces1148692671rocess @ A @ Z2 ) ) )
=> ( ! [Z1: proces634752977rocess @ A] :
( ? [Z2: proces634752977rocess @ A] :
( A2
= ( proces1915862579Choice @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E2 @ ( proces1148692671rocess @ A @ Z1 ) ) )
=> ~ ! [Z1: proces634752977rocess @ A,Z2: proces634752977rocess @ A] :
( ( A2
= ( proces1915862579Choice @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E2 @ ( proces1148692671rocess @ A @ Z2 ) ) ) ) ) ) ) ).
% process.set_cases
thf(fact_30_process_Oset__induct,axiom,
! [A: $tType,X: A,A2: proces634752977rocess @ A,P2: A > ( proces634752977rocess @ A ) > $o] :
( ( member @ A @ X @ ( proces1148692671rocess @ A @ A2 ) )
=> ( ! [Z1: A,Z2: proces634752977rocess @ A] : ( P2 @ Z1 @ ( proces1525233512Action @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: proces634752977rocess @ A,Xa: A] :
( ( member @ A @ Xa @ ( proces1148692671rocess @ A @ Z2 ) )
=> ( ( P2 @ Xa @ Z2 )
=> ( P2 @ Xa @ ( proces1525233512Action @ A @ Z1 @ Z2 ) ) ) )
=> ( ! [Z1a: proces634752977rocess @ A,Z2a: proces634752977rocess @ A,Xb: A] :
( ( member @ A @ Xb @ ( proces1148692671rocess @ A @ Z1a ) )
=> ( ( P2 @ Xb @ Z1a )
=> ( P2 @ Xb @ ( proces1915862579Choice @ A @ Z1a @ Z2a ) ) ) )
=> ( ! [Z1a: proces634752977rocess @ A,Z2a: proces634752977rocess @ A,Xc: A] :
( ( member @ A @ Xc @ ( proces1148692671rocess @ A @ Z2a ) )
=> ( ( P2 @ Xc @ Z2a )
=> ( P2 @ Xc @ ( proces1915862579Choice @ A @ Z1a @ Z2a ) ) ) )
=> ( P2 @ X @ A2 ) ) ) ) ) ) ).
% process.set_induct
thf(fact_31_process_Opred__inject_I2_J,axiom,
! [A: $tType,P2: A > $o,A2: proces634752977rocess @ A,Aa2: proces634752977rocess @ A] :
( ( proces1205983068rocess @ A @ P2 @ ( proces1915862579Choice @ A @ A2 @ Aa2 ) )
= ( ( proces1205983068rocess @ A @ P2 @ A2 )
& ( proces1205983068rocess @ A @ P2 @ Aa2 ) ) ) ).
% process.pred_inject(2)
thf(fact_32_process_Opred__inject_I1_J,axiom,
! [A: $tType,P2: A > $o,A2: A,Aa2: proces634752977rocess @ A] :
( ( proces1205983068rocess @ A @ P2 @ ( proces1525233512Action @ A @ A2 @ Aa2 ) )
= ( ( P2 @ A2 )
& ( proces1205983068rocess @ A @ P2 @ Aa2 ) ) ) ).
% process.pred_inject(1)
thf(fact_33_process_Osimps_I6_J,axiom,
! [B: $tType,A: $tType,F1: A > ( proces634752977rocess @ A ) > B,F2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces1406508781rocess @ A @ B @ F1 @ F2 @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% process.simps(6)
thf(fact_34_process_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: A > ( proces634752977rocess @ A ) > B,F2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,X11: A,X12: proces634752977rocess @ A] :
( ( proces1406508781rocess @ A @ B @ F1 @ F2 @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= ( F1 @ X11 @ X12 ) ) ).
% process.simps(5)
thf(fact_35_process_Ocollapse_I1_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( proces10484146Action @ A @ Process )
=> ( ( proces1525233512Action @ A @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) )
= Process ) ) ).
% process.collapse(1)
thf(fact_36_process_Ocollapse_I2_J,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( proces401113213Choice @ A @ Process )
=> ( ( proces1915862579Choice @ A @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) )
= Process ) ) ).
% process.collapse(2)
thf(fact_37_process_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( proces1406508781rocess @ A @ B )
= ( ^ [F12: A > ( proces634752977rocess @ A ) > B,F22: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,Process2: proces634752977rocess @ A] : ( if @ B @ ( proces10484146Action @ A @ Process2 ) @ ( F12 @ ( proces745025900prefOf @ A @ Process2 ) @ ( proces1778668539contOf @ A @ Process2 ) ) @ ( F22 @ ( proces979765041_ch1Of @ A @ Process2 ) @ ( proces988026546_ch2Of @ A @ Process2 ) ) ) ) ) ).
% process.case_eq_if
thf(fact_38_process_Opred__mono__strong,axiom,
! [A: $tType,P2: A > $o,X: proces634752977rocess @ A,Pa: A > $o] :
( ( proces1205983068rocess @ A @ P2 @ X )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( proces1148692671rocess @ A @ X ) )
=> ( ( P2 @ Z )
=> ( Pa @ Z ) ) )
=> ( proces1205983068rocess @ A @ Pa @ X ) ) ) ).
% process.pred_mono_strong
thf(fact_39_process_Opred__cong,axiom,
! [A: $tType,X: proces634752977rocess @ A,Ya: proces634752977rocess @ A,P2: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( proces1148692671rocess @ A @ Ya ) )
=> ( ( P2 @ Z )
= ( Pa @ Z ) ) )
=> ( ( proces1205983068rocess @ A @ P2 @ X )
= ( proces1205983068rocess @ A @ Pa @ Ya ) ) ) ) ).
% process.pred_cong
thf(fact_40_process_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: A > ( proces634752977rocess @ A ) > B,F2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,Process: proces634752977rocess @ A] :
( ( P2 @ ( proces1406508781rocess @ A @ B @ F1 @ F2 @ Process ) )
= ( ~ ( ( ( Process
= ( proces1525233512Action @ A @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) )
& ~ ( P2 @ ( F1 @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) ) )
| ( ( Process
= ( proces1915862579Choice @ A @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) )
& ~ ( P2 @ ( F2 @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) ) ) ) ) ) ).
% process.split_sel_asm
thf(fact_41_process_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: A > ( proces634752977rocess @ A ) > B,F2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > B,Process: proces634752977rocess @ A] :
( ( P2 @ ( proces1406508781rocess @ A @ B @ F1 @ F2 @ Process ) )
= ( ( ( Process
= ( proces1525233512Action @ A @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) )
=> ( P2 @ ( F1 @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) ) )
& ( ( Process
= ( proces1915862579Choice @ A @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) )
=> ( P2 @ ( F2 @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) ) ) ) ) ).
% process.split_sel
thf(fact_42_process_Oexhaust__sel,axiom,
! [A: $tType,Process: proces634752977rocess @ A] :
( ( Process
!= ( proces1525233512Action @ A @ ( proces745025900prefOf @ A @ Process ) @ ( proces1778668539contOf @ A @ Process ) ) )
=> ( Process
= ( proces1915862579Choice @ A @ ( proces979765041_ch1Of @ A @ Process ) @ ( proces988026546_ch2Of @ A @ Process ) ) ) ) ).
% process.exhaust_sel
thf(fact_43_process_Oexpand,axiom,
! [A: $tType,Process: proces634752977rocess @ A,Process3: proces634752977rocess @ A] :
( ( ( proces10484146Action @ A @ Process )
= ( proces10484146Action @ A @ Process3 ) )
=> ( ( ( proces10484146Action @ A @ Process )
=> ( ( proces10484146Action @ A @ Process3 )
=> ( ( ( proces745025900prefOf @ A @ Process )
= ( proces745025900prefOf @ A @ Process3 ) )
& ( ( proces1778668539contOf @ A @ Process )
= ( proces1778668539contOf @ A @ Process3 ) ) ) ) )
=> ( ( ( proces401113213Choice @ A @ Process )
=> ( ( proces401113213Choice @ A @ Process3 )
=> ( ( ( proces979765041_ch1Of @ A @ Process )
= ( proces979765041_ch1Of @ A @ Process3 ) )
& ( ( proces988026546_ch2Of @ A @ Process )
= ( proces988026546_ch2Of @ A @ Process3 ) ) ) ) )
=> ( Process = Process3 ) ) ) ) ).
% process.expand
thf(fact_44_process_Ocoinduct,axiom,
! [A: $tType,R: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > $o,Process: proces634752977rocess @ A,Process3: proces634752977rocess @ A] :
( ( R @ Process @ Process3 )
=> ( ! [Process4: proces634752977rocess @ A,Process5: proces634752977rocess @ A] :
( ( R @ Process4 @ Process5 )
=> ( ( ( proces10484146Action @ A @ Process4 )
= ( proces10484146Action @ A @ Process5 ) )
& ( ( proces10484146Action @ A @ Process4 )
=> ( ( proces10484146Action @ A @ Process5 )
=> ( ( ( proces745025900prefOf @ A @ Process4 )
= ( proces745025900prefOf @ A @ Process5 ) )
& ( R @ ( proces1778668539contOf @ A @ Process4 ) @ ( proces1778668539contOf @ A @ Process5 ) ) ) ) )
& ( ( proces401113213Choice @ A @ Process4 )
=> ( ( proces401113213Choice @ A @ Process5 )
=> ( ( R @ ( proces979765041_ch1Of @ A @ Process4 ) @ ( proces979765041_ch1Of @ A @ Process5 ) )
& ( R @ ( proces988026546_ch2Of @ A @ Process4 ) @ ( proces988026546_ch2Of @ A @ Process5 ) ) ) ) ) ) )
=> ( Process = Process3 ) ) ) ).
% process.coinduct
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_process_Ocoinduct__strong,axiom,
! [A: $tType,R: ( proces634752977rocess @ A ) > ( proces634752977rocess @ A ) > $o,Process: proces634752977rocess @ A,Process3: proces634752977rocess @ A] :
( ( R @ Process @ Process3 )
=> ( ! [Process4: proces634752977rocess @ A,Process5: proces634752977rocess @ A] :
( ( R @ Process4 @ Process5 )
=> ( ( ( proces10484146Action @ A @ Process4 )
= ( proces10484146Action @ A @ Process5 ) )
& ( ( proces10484146Action @ A @ Process4 )
=> ( ( proces10484146Action @ A @ Process5 )
=> ( ( ( proces745025900prefOf @ A @ Process4 )
= ( proces745025900prefOf @ A @ Process5 ) )
& ( ( R @ ( proces1778668539contOf @ A @ Process4 ) @ ( proces1778668539contOf @ A @ Process5 ) )
| ( ( proces1778668539contOf @ A @ Process4 )
= ( proces1778668539contOf @ A @ Process5 ) ) ) ) ) )
& ( ( proces401113213Choice @ A @ Process4 )
=> ( ( proces401113213Choice @ A @ Process5 )
=> ( ( ( R @ ( proces979765041_ch1Of @ A @ Process4 ) @ ( proces979765041_ch1Of @ A @ Process5 ) )
| ( ( proces979765041_ch1Of @ A @ Process4 )
= ( proces979765041_ch1Of @ A @ Process5 ) ) )
& ( ( R @ ( proces988026546_ch2Of @ A @ Process4 ) @ ( proces988026546_ch2Of @ A @ Process5 ) )
| ( ( proces988026546_ch2Of @ A @ Process4 )
= ( proces988026546_ch2Of @ A @ Process5 ) ) ) ) ) ) ) )
=> ( Process = Process3 ) ) ) ).
% process.coinduct_strong
thf(fact_50_process_Oset__sel_I2_J,axiom,
! [A: $tType,A2: proces634752977rocess @ A,X: A] :
( ( proces10484146Action @ A @ A2 )
=> ( ( member @ A @ X @ ( proces1148692671rocess @ A @ ( proces1778668539contOf @ A @ A2 ) ) )
=> ( member @ A @ X @ ( proces1148692671rocess @ A @ A2 ) ) ) ) ).
% process.set_sel(2)
thf(fact_51_process_Oset__sel_I1_J,axiom,
! [A: $tType,A2: proces634752977rocess @ A] :
( ( proces10484146Action @ A @ A2 )
=> ( member @ A @ ( proces745025900prefOf @ A @ A2 ) @ ( proces1148692671rocess @ A @ A2 ) ) ) ).
% process.set_sel(1)
thf(fact_52_process_Oset__sel_I4_J,axiom,
! [A: $tType,A2: proces634752977rocess @ A,Xb2: A] :
( ( proces401113213Choice @ A @ A2 )
=> ( ( member @ A @ Xb2 @ ( proces1148692671rocess @ A @ ( proces988026546_ch2Of @ A @ A2 ) ) )
=> ( member @ A @ Xb2 @ ( proces1148692671rocess @ A @ A2 ) ) ) ) ).
% process.set_sel(4)
thf(fact_53_process_Oset__sel_I3_J,axiom,
! [A: $tType,A2: proces634752977rocess @ A,Xa2: A] :
( ( proces401113213Choice @ A @ A2 )
=> ( ( member @ A @ Xa2 @ ( proces1148692671rocess @ A @ ( proces979765041_ch1Of @ A @ A2 ) ) )
=> ( member @ A @ Xa2 @ ( proces1148692671rocess @ A @ A2 ) ) ) ) ).
% process.set_sel(3)
thf(fact_54_process_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: proces634752977rocess @ A,Xa2: proces634752977rocess @ A,F: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( proces1148692671rocess @ A @ X ) )
=> ( ( member @ A @ Za @ ( proces1148692671rocess @ A @ Xa2 ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( proces370407749rocess @ A @ B @ F @ X )
= ( proces370407749rocess @ A @ B @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% process.inj_map_strong
thf(fact_55_process_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: proces634752977rocess @ A,F: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( proces1148692671rocess @ A @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( proces370407749rocess @ A @ B @ F @ X )
= ( proces370407749rocess @ A @ B @ G @ X ) ) ) ).
% process.map_cong0
thf(fact_56_process_Omap__cong,axiom,
! [B: $tType,A: $tType,X: proces634752977rocess @ A,Ya: proces634752977rocess @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( proces1148692671rocess @ A @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( proces370407749rocess @ A @ B @ F @ X )
= ( proces370407749rocess @ A @ B @ G @ Ya ) ) ) ) ).
% process.map_cong
thf(fact_57_process_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: proces634752977rocess @ A] : ( member @ A @ A1 @ ( proces1148692671rocess @ A @ ( proces1525233512Action @ A @ A1 @ A22 ) ) ) ).
% process.set_intros(1)
thf(fact_58_process_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: proces634752977rocess @ A,A1: A] :
( ( member @ A @ X @ ( proces1148692671rocess @ A @ A22 ) )
=> ( member @ A @ X @ ( proces1148692671rocess @ A @ ( proces1525233512Action @ A @ A1 @ A22 ) ) ) ) ).
% process.set_intros(2)
thf(fact_59_process_Oset__intros_I3_J,axiom,
! [A: $tType,Xa2: A,A1a: proces634752977rocess @ A,A2a: proces634752977rocess @ A] :
( ( member @ A @ Xa2 @ ( proces1148692671rocess @ A @ A1a ) )
=> ( member @ A @ Xa2 @ ( proces1148692671rocess @ A @ ( proces1915862579Choice @ A @ A1a @ A2a ) ) ) ) ).
% process.set_intros(3)
thf(fact_60_process_Oset__intros_I4_J,axiom,
! [A: $tType,Xb2: A,A2a: proces634752977rocess @ A,A1a: proces634752977rocess @ A] :
( ( member @ A @ Xb2 @ ( proces1148692671rocess @ A @ A2a ) )
=> ( member @ A @ Xb2 @ ( proces1148692671rocess @ A @ ( proces1915862579Choice @ A @ A1a @ A2a ) ) ) ) ).
% process.set_intros(4)
thf(fact_61_process_Osel_I2_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] :
( ( proces1778668539contOf @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= X12 ) ).
% process.sel(2)
thf(fact_62_process_Osel_I3_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces979765041_ch1Of @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= X21 ) ).
% process.sel(3)
thf(fact_63_process_Osel_I4_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces988026546_ch2Of @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= X22 ) ).
% process.sel(4)
thf(fact_64_process_Osel_I1_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] :
( ( proces745025900prefOf @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= X11 ) ).
% process.sel(1)
thf(fact_65_process_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,Q12: C > $o,G11: C > A,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C] :
( ( P @ A2 )
=> ( ( ( Q12 @ A2 )
=> ( ( proces1778668539contOf @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( G121 @ A2 ) ) )
& ( ~ ( Q12 @ A2 )
=> ( ( proces1778668539contOf @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ ( G122 @ A2 ) ) ) ) ) ) ).
% process.corec_sel(2)
thf(fact_66_process_Ocorec__sel_I3_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,Q21: C > $o,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C] :
( ~ ( P @ A2 )
=> ( ( ( Q21 @ A2 )
=> ( ( proces979765041_ch1Of @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( G211 @ A2 ) ) )
& ( ~ ( Q21 @ A2 )
=> ( ( proces979765041_ch1Of @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ ( G212 @ A2 ) ) ) ) ) ) ).
% process.corec_sel(3)
thf(fact_67_process_Ocorec__sel_I4_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,Q22: C > $o,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,G221: C > ( proces634752977rocess @ A ),G222: C > C] :
( ~ ( P @ A2 )
=> ( ( ( Q22 @ A2 )
=> ( ( proces988026546_ch2Of @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( G221 @ A2 ) ) )
& ( ~ ( Q22 @ A2 )
=> ( ( proces988026546_ch2Of @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ ( G222 @ A2 ) ) ) ) ) ) ).
% process.corec_sel(4)
thf(fact_68_process_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G11: C > A,Q12: C > $o,G121: C > ( proces634752977rocess @ A ),G122: C > C,Q21: C > $o,G211: C > ( proces634752977rocess @ A ),G212: C > C,Q22: C > $o,G221: C > ( proces634752977rocess @ A ),G222: C > C] :
( ( P @ A2 )
=> ( ( proces745025900prefOf @ A @ ( proces894737309rocess @ C @ A @ P @ G11 @ Q12 @ G121 @ G122 @ Q21 @ G211 @ G212 @ Q22 @ G221 @ G222 @ A2 ) )
= ( G11 @ A2 ) ) ) ).
% process.corec_sel(1)
thf(fact_69_process_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P2: ( proces634752977rocess @ A ) > ( proces634752977rocess @ B ) > $o,X: proces634752977rocess @ A,Y: proces634752977rocess @ B,R: A > B > $o] :
( ( P2 @ X @ Y )
=> ( ! [Process4: proces634752977rocess @ A,Process5: proces634752977rocess @ B] :
( ( P2 @ Process4 @ Process5 )
=> ( ( ( proces10484146Action @ A @ Process4 )
= ( proces10484146Action @ B @ Process5 ) )
& ( ( proces10484146Action @ A @ Process4 )
=> ( ( proces10484146Action @ B @ Process5 )
=> ( ( R @ ( proces745025900prefOf @ A @ Process4 ) @ ( proces745025900prefOf @ B @ Process5 ) )
& ( P2 @ ( proces1778668539contOf @ A @ Process4 ) @ ( proces1778668539contOf @ B @ Process5 ) ) ) ) )
& ( ( proces401113213Choice @ A @ Process4 )
=> ( ( proces401113213Choice @ B @ Process5 )
=> ( ( P2 @ ( proces979765041_ch1Of @ A @ Process4 ) @ ( proces979765041_ch1Of @ B @ Process5 ) )
& ( P2 @ ( proces988026546_ch2Of @ A @ Process4 ) @ ( proces988026546_ch2Of @ B @ Process5 ) ) ) ) ) ) )
=> ( proces749077512rocess @ A @ B @ R @ X @ Y ) ) ) ).
% process.rel_coinduct
thf(fact_70_process_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( proces749077512rocess @ A @ B )
= ( ^ [R2: A > B > $o,A4: proces634752977rocess @ A,B2: proces634752977rocess @ B] :
( ( ( proces10484146Action @ A @ A4 )
= ( proces10484146Action @ B @ B2 ) )
& ( ( proces10484146Action @ A @ A4 )
=> ( ( proces10484146Action @ B @ B2 )
=> ( ( R2 @ ( proces745025900prefOf @ A @ A4 ) @ ( proces745025900prefOf @ B @ B2 ) )
& ( proces749077512rocess @ A @ B @ R2 @ ( proces1778668539contOf @ A @ A4 ) @ ( proces1778668539contOf @ B @ B2 ) ) ) ) )
& ( ( proces401113213Choice @ A @ A4 )
=> ( ( proces401113213Choice @ B @ B2 )
=> ( ( proces749077512rocess @ A @ B @ R2 @ ( proces979765041_ch1Of @ A @ A4 ) @ ( proces979765041_ch1Of @ B @ B2 ) )
& ( proces749077512rocess @ A @ B @ R2 @ ( proces988026546_ch2Of @ A @ A4 ) @ ( proces988026546_ch2Of @ B @ B2 ) ) ) ) ) ) ) ) ).
% process.rel_sel
thf(fact_71_process_Osimps_I21_J,axiom,
! [A: $tType,X11: A,X12: proces634752977rocess @ A] :
( ( proces1148692671rocess @ A @ ( proces1525233512Action @ A @ X11 @ X12 ) )
= ( insert @ A @ X11 @ ( proces1148692671rocess @ A @ X12 ) ) ) ).
% process.simps(21)
thf(fact_72_process_Osimps_I22_J,axiom,
! [A: $tType,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A] :
( ( proces1148692671rocess @ A @ ( proces1915862579Choice @ A @ X21 @ X22 ) )
= ( sup_sup @ ( set @ A ) @ ( proces1148692671rocess @ A @ X21 ) @ ( proces1148692671rocess @ A @ X22 ) ) ) ).
% process.simps(22)
thf(fact_73_process_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: proces634752977rocess @ A] :
( ( proces1205983068rocess @ B @ Q @ ( proces370407749rocess @ A @ B @ F @ X ) )
= ( proces1205983068rocess @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% process.pred_map
thf(fact_74_process_Opred__set,axiom,
! [A: $tType] :
( ( proces1205983068rocess @ A )
= ( ^ [P3: A > $o,X2: proces634752977rocess @ A] :
! [Y2: A] :
( ( member @ A @ Y2 @ ( proces1148692671rocess @ A @ X2 ) )
=> ( P3 @ Y2 ) ) ) ) ).
% process.pred_set
thf(fact_75_process_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: proces634752977rocess @ A] :
( ( proces1148692671rocess @ B @ ( proces370407749rocess @ A @ B @ F @ V ) )
= ( image @ A @ B @ F @ ( proces1148692671rocess @ A @ V ) ) ) ).
% process.set_map
thf(fact_76_process_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: proces634752977rocess @ A,B3: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R @ A2 @ B3 )
=> ( ! [X1: A,X23: proces634752977rocess @ A] :
( ( A2
= ( proces1525233512Action @ A @ X1 @ X23 ) )
=> ! [Y1: B,Y23: proces634752977rocess @ B] :
( ( B3
= ( proces1525233512Action @ B @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( proces749077512rocess @ A @ B @ R @ X23 @ Y23 ) ) ) )
=> ~ ! [X1a: proces634752977rocess @ A,X2a: proces634752977rocess @ A] :
( ( A2
= ( proces1915862579Choice @ A @ X1a @ X2a ) )
=> ! [Y1a: proces634752977rocess @ B,Y2a: proces634752977rocess @ B] :
( ( B3
= ( proces1915862579Choice @ B @ Y1a @ Y2a ) )
=> ( ( proces749077512rocess @ A @ B @ R @ X1a @ Y1a )
=> ~ ( proces749077512rocess @ A @ B @ R @ X2a @ Y2a ) ) ) ) ) ) ).
% process.rel_cases
thf(fact_77_process_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X11: A,X12: proces634752977rocess @ A,Y21: proces634752977rocess @ B,Y22: proces634752977rocess @ B] :
~ ( proces749077512rocess @ A @ B @ R @ ( proces1525233512Action @ A @ X11 @ X12 ) @ ( proces1915862579Choice @ B @ Y21 @ Y22 ) ) ).
% process.rel_distinct(1)
thf(fact_78_process_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: proces634752977rocess @ A,X22: proces634752977rocess @ A,Y21: proces634752977rocess @ B,Y22: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R @ ( proces1915862579Choice @ A @ X21 @ X22 ) @ ( proces1915862579Choice @ B @ Y21 @ Y22 ) )
= ( ( proces749077512rocess @ A @ B @ R @ X21 @ Y21 )
& ( proces749077512rocess @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% process.rel_inject(2)
thf(fact_79_process_Orel__inject_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X11: A,X12: proces634752977rocess @ A,Y11: B,Y12: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R @ ( proces1525233512Action @ A @ X11 @ X12 ) @ ( proces1525233512Action @ B @ Y11 @ Y12 ) )
= ( ( R @ X11 @ Y11 )
& ( proces749077512rocess @ A @ B @ R @ X12 @ Y12 ) ) ) ).
% process.rel_inject(1)
thf(fact_80_process_Orel__eq,axiom,
! [A: $tType] :
( ( proces749077512rocess @ A @ A
@ ^ [Y3: A,Z3: A] : Y3 = Z3 )
= ( ^ [Y3: proces634752977rocess @ A,Z3: proces634752977rocess @ A] : Y3 = Z3 ) ) ).
% process.rel_eq
thf(fact_81_process_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: proces634752977rocess @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( proces749077512rocess @ B @ B @ Ra @ X @ X ) ) ).
% process.rel_refl
thf(fact_82_process_Omap__o__corec,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > $o,Ga: C > A,Gb: C > $o,Gc: C > ( proces634752977rocess @ A ),Gd: C > C,Ge: C > $o,Gf: C > ( proces634752977rocess @ A ),Gg: C > C,Gh: C > $o,Gi: C > ( proces634752977rocess @ A ),Gj: C > C] :
( ( comp @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ C @ ( proces370407749rocess @ A @ B @ F ) @ ( proces894737309rocess @ C @ A @ G @ Ga @ Gb @ Gc @ Gd @ Ge @ Gf @ Gg @ Gh @ Gi @ Gj ) )
= ( proces894737309rocess @ C @ B @ G @ ( comp @ A @ B @ C @ F @ Ga ) @ Gb @ ( comp @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ C @ ( proces370407749rocess @ A @ B @ F ) @ Gc ) @ Gd @ Ge @ ( comp @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ C @ ( proces370407749rocess @ A @ B @ F ) @ Gf ) @ Gg @ Gh @ ( comp @ ( proces634752977rocess @ A ) @ ( proces634752977rocess @ B ) @ C @ ( proces370407749rocess @ A @ B @ F ) @ Gi ) @ Gj ) ) ).
% process.map_o_corec
thf(fact_83_process_Orel__cong,axiom,
! [A: $tType,B: $tType,X: proces634752977rocess @ A,Ya: proces634752977rocess @ A,Y: proces634752977rocess @ B,Xa2: proces634752977rocess @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( proces1148692671rocess @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( proces1148692671rocess @ B @ Xa2 ) )
=> ( ( R @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( proces749077512rocess @ A @ B @ R @ X @ Y )
= ( proces749077512rocess @ A @ B @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% process.rel_cong
thf(fact_84_process_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: proces634752977rocess @ A,Y: proces634752977rocess @ B,Ra: A > B > $o] :
( ( proces749077512rocess @ A @ B @ R @ X @ Y )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( proces1148692671rocess @ A @ X ) )
=> ( ( member @ B @ Yb @ ( proces1148692671rocess @ B @ Y ) )
=> ( ( R @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( proces749077512rocess @ A @ B @ Ra @ X @ Y ) ) ) ).
% process.rel_mono_strong
thf(fact_85_process_Orel__refl__strong,axiom,
! [A: $tType,X: proces634752977rocess @ A,Ra: A > A > $o] :
( ! [Z: A] :
( ( member @ A @ Z @ ( proces1148692671rocess @ A @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( proces749077512rocess @ A @ A @ Ra @ X @ X ) ) ).
% process.rel_refl_strong
thf(fact_86_process_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: proces634752977rocess @ A,Y21: proces634752977rocess @ B,X22: proces634752977rocess @ A,Y22: proces634752977rocess @ B] :
( ( proces749077512rocess @ A @ B @ R @ X21 @ Y21 )
=> ( ( proces749077512rocess @ A @ B @ R @ X22 @ Y22 )
=> ( proces749077512rocess @ A @ B @ R @ ( proces1915862579Choice @ A @ X21 @ X22 ) @ ( proces1915862579Choice @ B @ Y21 @ Y22 ) ) ) ) ).
% process.rel_intros(2)
thf(fact_87_process_Orel__intros_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X11: A,Y11: B,X12: proces634752977rocess @ A,Y12: proces634752977rocess @ B] :
( ( R @ X11 @ Y11 )
=> ( ( proces749077512rocess @ A @ B @ R @ X12 @ Y12 )
=> ( proces749077512rocess @ A @ B @ R @ ( proces1525233512Action @ A @ X11 @ X12 ) @ ( proces1525233512Action @ B @ Y11 @ Y12 ) ) ) ) ).
% process.rel_intros(1)
thf(fact_88_process_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: proces634752977rocess @ A] :
( ( proces370407749rocess @ B @ C @ G @ ( proces370407749rocess @ A @ B @ F @ V ) )
= ( proces370407749rocess @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% process.map_comp
thf(fact_89_process_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: proces634752977rocess @ A,Y22: proces634752977rocess @ A,X11: B,X12: proces634752977rocess @ B] :
~ ( proces749077512rocess @ A @ B @ R @ ( proces1915862579Choice @ A @ Y21 @ Y22 ) @ ( proces1525233512Action @ B @ X11 @ X12 ) ) ).
% process.rel_distinct(2)
thf(fact_90_Un__insert__left,axiom,
! [A: $tType,A2: A,B4: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B4 ) @ C2 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_91_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A2: A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B4 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_92_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A2: B,B4: set @ B] :
( ( image @ B @ A @ F @ ( insert @ B @ A2 @ B4 ) )
= ( insert @ A @ ( F @ A2 ) @ ( image @ B @ A @ F @ B4 ) ) ) ).
% image_insert
thf(fact_93_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( insert @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) )
= ( image @ A @ B @ F @ A3 ) ) ) ).
% insert_image
thf(fact_94_Ball__image__comp,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,G: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F @ A3 ) )
=> ( G @ X2 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ( comp @ A @ $o @ B @ G @ F @ X2 ) ) ) ) ).
% Ball_image_comp
thf(fact_95_UnCI,axiom,
! [A: $tType,C3: A,B4: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C3 @ B4 )
=> ( member @ A @ C3 @ A3 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_96_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X: B,A3: set @ B] :
( ( B3
= ( F @ X ) )
=> ( ( member @ B @ X @ A3 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_97_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
= ( insert @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_98_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_99_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_100_Un__iff,axiom,
! [A: $tType,C3: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
= ( ( member @ A @ C3 @ A3 )
| ( member @ A @ C3 @ B4 ) ) ) ).
% Un_iff
thf(fact_101_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,B3: B,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( ( B3
= ( F @ X ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_102_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P2: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
=> ( P2 @ X3 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_103_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_104_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P2: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
& ( P2 @ X4 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A3 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_105_image__iff,axiom,
! [A: $tType,B: $tType,Z4: A,F: B > A,A3: set @ B] :
( ( member @ A @ Z4 @ ( image @ B @ A @ F @ A3 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( Z4
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_106_imageI,axiom,
! [B: $tType,A: $tType,X: A,A3: set @ A,F: A > B] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F @ X ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% imageI
thf(fact_107_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B5: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B5 ) )
& ~ ( member @ A @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_108_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A3 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_109_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B3 @ B4 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A3 = B4 ) )
& ( ( A2 != B3 )
=> ? [C4: set @ A] :
( ( A3
= ( insert @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B4
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_110_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_111_insert__ident,axiom,
! [A: $tType,X: A,A3: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ~ ( member @ A @ X @ B4 )
=> ( ( ( insert @ A @ X @ A3 )
= ( insert @ A @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_112_Set_Oset__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ~ ! [B5: set @ A] :
( ( A3
= ( insert @ A @ X @ B5 ) )
=> ( member @ A @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_113_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( member @ A @ A2 @ B4 )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_114_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_115_insertE,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_116_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_117_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A5: set @ A,P3: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( P3 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_118_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B4 @ C2 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_119_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_120_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_121_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_122_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) @ C2 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_123_ball__Un,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,P2: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P2 @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_124_bex__Un,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,P2: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P2 @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B4 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_125_UnI2,axiom,
! [A: $tType,C3: A,B4: set @ A,A3: set @ A] :
( ( member @ A @ C3 @ B4 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_126_UnI1,axiom,
! [A: $tType,C3: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C3 @ A3 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_127_UnE,axiom,
! [A: $tType,C3: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
=> ( ~ ( member @ A @ C3 @ A3 )
=> ( member @ A @ C3 @ B4 ) ) ) ).
% UnE
thf(fact_128_image__Un,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,B4: set @ B] :
( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A3 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) @ ( image @ B @ A @ F @ B4 ) ) ) ).
% image_Un
thf(fact_129_not__False__in__image__Ball,axiom,
! [A: $tType,P2: A > $o,A3: set @ A] :
( ( ~ ( member @ $o @ $false @ ( image @ A @ $o @ P2 @ A3 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( P2 @ X2 ) ) ) ) ).
% not_False_in_image_Ball
thf(fact_130_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F3: B > A,G2: C > B,X2: C] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_131_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_132_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_133_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_134_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_135_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_136_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_137_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B4: set @ B,C2: B > A,D2: B > A,Inf: ( set @ A ) > A] :
( ( A3 = B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C2 @ A3 ) )
= ( Inf @ ( image @ B @ A @ D2 @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_138_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B4: set @ B,C2: B > A,D2: B > A,Sup: ( set @ A ) > A] :
( ( A3 = B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C2 @ A3 ) )
= ( Sup @ ( image @ B @ A @ D2 @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_139_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_140_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z4 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_141_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_142_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_143_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_144_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C3 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.assoc
thf(fact_145_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z4 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) ) ) ) ).
% sup_assoc
thf(fact_146_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A4: A,B2: A] : ( sup_sup @ A @ B2 @ A4 ) ) ) ) ).
% sup.commute
thf(fact_147_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_148_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C3 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C3 ) ) ) ) ).
% sup.left_commute
thf(fact_149_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z4 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z4 ) ) ) ) ).
% sup_left_commute
thf(fact_150_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F3: B > C,G2: A > B,X2: A] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_151_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_152_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B3: A > C,C3: D > B,D3: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B3 )
= ( comp @ D @ B @ A @ C3 @ D3 ) )
=> ( ( A2 @ ( B3 @ V ) )
= ( C3 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_153_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B3: A > C,C3: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B3 )
= ( comp @ D @ B @ A @ C3 @ D3 ) )
=> ! [V2: A] :
( ( A2 @ ( B3 @ V2 ) )
= ( C3 @ ( D3 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_154_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B3: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B3 )
= C3 )
=> ( ( A2 @ ( B3 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_155_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R3: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R3 ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_156_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A3: set @ B,G: C > A,B4: set @ C,H: A > D] :
( ( ( image @ B @ A @ F @ A3 )
= ( image @ C @ A @ G @ B4 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A3 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_157_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A3: set @ C] :
( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A3 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_158_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A3: set @ C] :
( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A3 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_159_ball__reg,axiom,
! [A: $tType,R: set @ A,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ R )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ R )
=> ( P2 @ X3 ) )
=> ! [X4: A] :
( ( member @ A @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_160_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F4: D > A,G3: E > D,X5: E] :
( ( ( F @ ( G @ X ) )
= ( F4 @ ( G3 @ X5 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ E @ F4 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_161_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_162_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D > A,H: C > D,F: A > E] :
( ( ( M @ ( G @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_163_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R22: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D @ B @ A @ R1 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
= ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_164_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R3: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R3 )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
= ( comp @ D @ B @ E @ L1 @ R3 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_165_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R3: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H )
= R3 )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
= ( comp @ B @ D @ A @ F @ R3 ) ) ) ).
% rewriteR_comp_comp
thf(fact_166_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_167_process_Opred__rel,axiom,
! [A: $tType] :
( ( proces1205983068rocess @ A )
= ( ^ [P3: A > $o,X2: proces634752977rocess @ A] : ( proces749077512rocess @ A @ A @ ( bNF_eq_onp @ A @ P3 ) @ X2 @ X2 ) ) ) ).
% process.pred_rel
thf(fact_168_Sup__insert,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A2: A,A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ A3 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_insert
thf(fact_169_SUP__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [G: B > A,F: C > B,A3: set @ C] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A3 ) ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A3 ) ) ) ) ).
% SUP_image
thf(fact_170_Union__Un__distrib,axiom,
! [A: $tType,A3: set @ ( set @ A ),B4: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ A3 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) ) ) ).
% Union_Un_distrib
thf(fact_171_SUP__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [A3: set @ B,B4: set @ B,C2: B > A,D2: B > A] :
( ( A3 = B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C2 @ A3 ) )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ D2 @ B4 ) ) ) ) ) ) ).
% SUP_cong
thf(fact_172_process_Orel__eq__onp,axiom,
! [A: $tType,P2: A > $o] :
( ( proces749077512rocess @ A @ A @ ( bNF_eq_onp @ A @ P2 ) )
= ( bNF_eq_onp @ ( proces634752977rocess @ A ) @ ( proces1205983068rocess @ A @ P2 ) ) ) ).
% process.rel_eq_onp
thf(fact_173_eq__onp__eqD,axiom,
! [A: $tType,P2: A > $o,Q: A > A > $o,X: A] :
( ( ( bNF_eq_onp @ A @ P2 )
= Q )
=> ( ( P2 @ X )
= ( Q @ X @ X ) ) ) ).
% eq_onp_eqD
thf(fact_174_eq__onp__mono0,axiom,
! [A: $tType,A3: set @ A,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ A3 )
=> ( ( bNF_eq_onp @ A @ P2 @ X4 @ Xa3 )
=> ( bNF_eq_onp @ A @ Q @ X4 @ Xa3 ) ) ) ) ) ).
% eq_onp_mono0
thf(fact_175_eq__onp__to__eq,axiom,
! [A: $tType,P2: A > $o,X: A,Y: A] :
( ( bNF_eq_onp @ A @ P2 @ X @ Y )
=> ( X = Y ) ) ).
% eq_onp_to_eq
thf(fact_176_eq__onp__same__args,axiom,
! [A: $tType,P2: A > $o,X: A] :
( ( bNF_eq_onp @ A @ P2 @ X @ X )
= ( P2 @ X ) ) ).
% eq_onp_same_args
thf(fact_177_Union__insert,axiom,
! [A: $tType,A2: set @ A,B4: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( insert @ ( set @ A ) @ A2 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) ) ) ).
% Union_insert
thf(fact_178_Sup__union__distrib,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A3: set @ A,B4: set @ A] :
( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ).
% Sup_union_distrib
thf(fact_179_Union__image__insert,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A2: B,B4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ ( insert @ B @ A2 @ B4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ B4 ) ) ) ) ).
% Union_image_insert
thf(fact_180_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F5: $tType,A: $tType,C: $tType,B: $tType,G4: $tType,F1: F5 > A,F2: D > F5,G1: G4 > B,G22: E > G4] :
( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F5 @ A @ D @ F1 @ F2 ) @ ( comp @ G4 @ B @ E @ G1 @ G22 ) )
= ( comp @ ( F5 > G4 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F5 @ E @ G4 @ C @ F2 @ G22 ) @ ( bNF_vimage2p @ F5 @ A @ G4 @ B @ C @ F1 @ G1 ) ) ) ).
% vimage2p_comp
thf(fact_181_INF__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Inf @ A @ ( type2 @ A ) )
=> ! [G: B > A,F: C > B,A3: set @ C] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A3 ) ) )
= ( complete_Inf_Inf @ A @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A3 ) ) ) ) ).
% INF_image
thf(fact_182_SUP__id__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( id @ A ) @ A3 ) )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ).
% SUP_id_eq
thf(fact_183_UN__ball__bex__simps_I3_J,axiom,
! [D: $tType,A3: set @ ( set @ D ),P2: D > $o] :
( ( ? [X2: D] :
( ( member @ D @ X2 @ ( complete_Sup_Sup @ ( set @ D ) @ A3 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: set @ D] :
( ( member @ ( set @ D ) @ X2 @ A3 )
& ? [Y2: D] :
( ( member @ D @ Y2 @ X2 )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_184_UN__ball__bex__simps_I1_J,axiom,
! [A: $tType,A3: set @ ( set @ A ),P2: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A3 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ X2 )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_185_UnionI,axiom,
! [A: $tType,X6: set @ A,C2: set @ ( set @ A ),A3: A] :
( ( member @ ( set @ A ) @ X6 @ C2 )
=> ( ( member @ A @ A3 @ X6 )
=> ( member @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) ) ) ) ).
% UnionI
thf(fact_186_Union__iff,axiom,
! [A: $tType,A3: A,C2: set @ ( set @ A )] :
( ( member @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
= ( ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C2 )
& ( member @ A @ A3 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_187_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X2: A] : X2 ) ) ).
% id_apply
thf(fact_188_image__id,axiom,
! [A: $tType] :
( ( image @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% image_id
thf(fact_189_ball__UN,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A3: set @ B,P2: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B4 @ A3 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ ( B4 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_190_bex__UN,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A3: set @ B,P2: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B4 @ A3 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ? [Y2: A] :
( ( member @ A @ Y2 @ ( B4 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_191_UN__ball__bex__simps_I2_J,axiom,
! [C: $tType,B: $tType,B4: B > ( set @ C ),A3: set @ B,P2: C > $o] :
( ( ! [X2: C] :
( ( member @ C @ X2 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B4 @ A3 ) ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A3 )
=> ! [Y2: C] :
( ( member @ C @ Y2 @ ( B4 @ X2 ) )
=> ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_192_UN__ball__bex__simps_I4_J,axiom,
! [F5: $tType,E: $tType,B4: E > ( set @ F5 ),A3: set @ E,P2: F5 > $o] :
( ( ? [X2: F5] :
( ( member @ F5 @ X2 @ ( complete_Sup_Sup @ ( set @ F5 ) @ ( image @ E @ ( set @ F5 ) @ B4 @ A3 ) ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: E] :
( ( member @ E @ X2 @ A3 )
& ? [Y2: F5] :
( ( member @ F5 @ Y2 @ ( B4 @ X2 ) )
& ( P2 @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_193_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T )
= T ) ).
% fun.map_id
thf(fact_194_id__comp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G )
= G ) ).
% id_comp
thf(fact_195_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_196_INF__id__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] :
( ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( id @ A ) @ A3 ) )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ).
% INF_id_eq
thf(fact_197_UnionE,axiom,
! [A: $tType,A3: A,C2: set @ ( set @ A )] :
( ( member @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ C2 ) )
=> ~ ! [X7: set @ A] :
( ( member @ A @ A3 @ X7 )
=> ~ ( member @ ( set @ A ) @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_198_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_199_process_Omap__id0,axiom,
! [A: $tType] :
( ( proces370407749rocess @ A @ A @ ( id @ A ) )
= ( id @ ( proces634752977rocess @ A ) ) ) ).
% process.map_id0
thf(fact_200_process_Omap__id,axiom,
! [A: $tType,T: proces634752977rocess @ A] :
( ( proces370407749rocess @ A @ A @ ( id @ A ) @ T )
= T ) ).
% process.map_id
thf(fact_201_Inf_OINF__id__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
( ( Inf @ ( image @ A @ A @ ( id @ A ) @ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_id_eq
thf(fact_202_Sup_OSUP__id__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
( ( Sup @ ( image @ A @ A @ ( id @ A ) @ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_id_eq
thf(fact_203_INF__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Inf @ A @ ( type2 @ A ) )
=> ! [A3: set @ B,B4: set @ B,C2: B > A,D2: B > A] :
( ( A3 = B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ C2 @ A3 ) )
= ( complete_Inf_Inf @ A @ ( image @ B @ A @ D2 @ B4 ) ) ) ) ) ) ).
% INF_cong
thf(fact_204_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X2: A] :
( ( F @ X2 )
= X2 ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_205_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X2: A] : X2 ) ) ).
% id_def
thf(fact_206_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B3: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B3 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C3 ) )
=> ( ( A2 @ ( B3 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_207_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X ) )
= X ) ) ).
% pointfree_idE
thf(fact_208_member__bind,axiom,
! [A: $tType,B: $tType,X: A,P2: set @ B,F: B > ( set @ A )] :
( ( member @ A @ X @ ( bind @ B @ A @ P2 @ F ) )
= ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ P2 ) ) ) ) ).
% member_bind
thf(fact_209_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S: set @ ( A > $o ),X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_210_Union__image__empty,axiom,
! [B: $tType,A: $tType,A3: set @ A,F: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ ( bot_bot @ ( set @ B ) ) ) ) )
= A3 ) ).
% Union_image_empty
thf(fact_211_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_212_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_213_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_214_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X2: A] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_215_Sup1__I,axiom,
! [A: $tType,P2: A > $o,A3: set @ ( A > $o ),A2: A] :
( ( member @ ( A > $o ) @ P2 @ A3 )
=> ( ( P2 @ A2 )
=> ( complete_Sup_Sup @ ( A > $o ) @ A3 @ A2 ) ) ) ).
% Sup1_I
thf(fact_216_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_217_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image @ B @ A @ F @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_218_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_219_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_220_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_221_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_222_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_223_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( X2
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_224_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( X2
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_225_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_226_ball__empty,axiom,
! [A: $tType,P2: A > $o,X4: A] :
( ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
=> ( P2 @ X4 ) ) ).
% ball_empty
thf(fact_227_Un__empty,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_228_empty__bind,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A )] :
( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_229_Sup__empty,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_230_Sup1__E,axiom,
! [A: $tType,A3: set @ ( A > $o ),A2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ A3 @ A2 )
=> ~ ! [P4: A > $o] :
( ( member @ ( A > $o ) @ P4 @ A3 )
=> ~ ( P4 @ A2 ) ) ) ).
% Sup1_E
thf(fact_231_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_232_Union__empty__conv,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A3 )
=> ( X2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_233_empty__Union__conv,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A3 )
=> ( X2
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_234_sup__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_235_sup__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_236_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_237_Un__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_238_image__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,A3: set @ C,G: C > ( set @ B )] :
( ( image @ B @ A @ F @ ( bind @ C @ B @ A3 @ G ) )
= ( bind @ C @ A @ A3 @ ( comp @ ( set @ B ) @ ( set @ A ) @ C @ ( image @ B @ A @ F ) @ G ) ) ) ).
% image_bind
thf(fact_239_bind__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,A3: set @ C,G: B > ( set @ A )] :
( ( bind @ B @ A @ ( image @ C @ B @ F @ A3 ) @ G )
= ( bind @ C @ A @ A3 @ ( comp @ B @ ( set @ A ) @ C @ G @ F ) ) ) ).
% bind_image
thf(fact_240_singletonD,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_241_singleton__iff,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_242_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B3: A,C3: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B3 = D3 ) )
| ( ( A2 = D3 )
& ( B3 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_243_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_244_singleton__inject,axiom,
! [A: $tType,A2: A,B3: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_245_Inf__INT__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( A > $o ) )
= ( ^ [S: set @ ( A > $o ),X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_246_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_247_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_248_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_249_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_250_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [I: set @ B,F: B > A,X: A] :
( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I )
=> ( ( F @ I2 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_251_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [I: set @ B,F: B > A,X: A] :
( ( I
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I )
=> ( ( F @ I2 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ I ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_252_singleton__Un__iff,axiom,
! [A: $tType,X: A,A3: set @ A,B4: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B4 ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_253_Un__singleton__iff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B4 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_254_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A4: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_255_bind__UNION,axiom,
! [A: $tType,B: $tType] :
( ( bind @ B @ A )
= ( ^ [A5: set @ B,F3: B > ( set @ A )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F3 @ A5 ) ) ) ) ).
% bind_UNION
%----Type constructors (24)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounde1808546759up_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A6: $tType,A7: $tType] :
( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
=> ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A6: $tType,A7: $tType] :
( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
=> ( semilattice_sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A6: $tType,A7: $tType] :
( ( complete_Sup @ A7 @ ( type2 @ A7 ) )
=> ( complete_Sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A6: $tType,A7: $tType] :
( ( complete_Inf @ A7 @ ( type2 @ A7 ) )
=> ( complete_Inf @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A6: $tType,A7: $tType] :
( ( lattice @ A7 @ ( type2 @ A7 ) )
=> ( lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
! [A6: $tType] : ( bounde1808546759up_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_4,axiom,
! [A6: $tType] : ( comple187826305attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_5,axiom,
! [A6: $tType] : ( bounded_lattice_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_6,axiom,
! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_7,axiom,
! [A6: $tType] : ( complete_Sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_8,axiom,
! [A6: $tType] : ( complete_Inf @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_9,axiom,
! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_10,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_11,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_12,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_13,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_14,axiom,
complete_Sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_15,axiom,
complete_Inf @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_16,axiom,
lattice @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( proces10484146Action @ a @ p )
| ( proces401113213Choice @ a @ p ) ) ).
%------------------------------------------------------------------------------