TPTP Problem File: ITP004^5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP004^5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Epred__set_2EREST__SUBSET.p, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Epred__set_2EREST__SUBSET.p [Gau20]
% : HL401501^5.p [TPAP]
% Status : Theorem
% Rating : 0.62 v9.0.0, 0.70 v8.2.0, 0.85 v8.1.0, 0.73 v7.5.0
% Syntax : Number of formulae : 2697 ( 284 unt; 441 typ; 0 def)
% Number of atoms : 35934 (1626 equ; 0 cnn)
% Maximal formula atoms : 160 ( 15 avg)
% Number of connectives : 62196 ( 292 ~; 236 |;1258 &;55803 @)
% ( 813 <=>;3794 =>; 0 <=; 0 <~>)
% Maximal formula depth : 42 ( 10 avg)
% Number of types : 10 ( 8 usr)
% Number of type conns : 457 ( 457 >; 0 *; 0 +; 0 <<)
% Number of symbols : 461 ( 458 usr; 149 con; 0-4 aty)
% Number of variables : 6881 ( 339 ^;6300 !; 242 ?;6881 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
include('Axioms/ITP001/ITP002^5.ax').
include('Axioms/ITP001/ITP003^5.ax').
include('Axioms/ITP001/ITP004^5.ax').
include('Axioms/ITP001/ITP007^5.ax').
include('Axioms/ITP001/ITP006^5.ax').
include('Axioms/ITP001/ITP005^5.ax').
include('Axioms/ITP001/ITP008^5.ax').
include('Axioms/ITP001/ITP009^5.ax').
include('Axioms/ITP001/ITP010^5.ax').
include('Axioms/ITP001/ITP012^5.ax').
include('Axioms/ITP001/ITP011^5.ax').
include('Axioms/ITP001/ITP013^5.ax').
include('Axioms/ITP001/ITP014^5.ax').
include('Axioms/ITP001/ITP015^5.ax').
include('Axioms/ITP001/ITP017^5.ax').
include('Axioms/ITP001/ITP016^5.ax').
include('Axioms/ITP001/ITP019^5.ax').
include('Axioms/ITP001/ITP018^5.ax').
include('Axioms/ITP001/ITP021^5.ax').
include('Axioms/ITP001/ITP022^5.ax').
include('Axioms/ITP001/ITP020^5.ax').
include('Axioms/ITP001/ITP024^5.ax').
include('Axioms/ITP001/ITP023^5.ax').
include('Axioms/ITP001/ITP025^5.ax').
include('Axioms/ITP001/ITP026^5.ax').
%------------------------------------------------------------------------------
thf(tp_c_2Epred__set_2EBIGINTER,type,
c_2Epred__set_2EBIGINTER: del > $i ).
thf(mem_c_2Epred__set_2EBIGINTER,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION: del > $i ).
thf(mem_c_2Epred__set_2EBIGUNION,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Epred__set_2EBIJ,type,
c_2Epred__set_2EBIJ: del > del > $i ).
thf(mem_c_2Epred__set_2EBIJ,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD: del > $i ).
thf(mem_c_2Epred__set_2ECARD,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE: del > $i ).
thf(mem_c_2Epred__set_2ECHOICE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ A_27a ) ) ).
thf(tp_c_2Epred__set_2ECOMPL,type,
c_2Epred__set_2ECOMPL: del > $i ).
thf(mem_c_2Epred__set_2ECOMPL,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Epred__set_2ECROSS,type,
c_2Epred__set_2ECROSS: del > del > $i ).
thf(mem_c_2Epred__set_2ECROSS,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE: del > $i ).
thf(mem_c_2Epred__set_2EDELETE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EDFUNSET,type,
c_2Epred__set_2EDFUNSET: del > del > $i ).
thf(mem_c_2Epred__set_2EDFUNSET,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF: del > $i ).
thf(mem_c_2Epred__set_2EDIFF,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT: del > $i ).
thf(mem_c_2Epred__set_2EDISJOINT,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY: del > $i ).
thf(mem_c_2Epred__set_2EEMPTY,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ ( arr @ A_27a @ bool ) ) ).
thf(tp_c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE: del > $i ).
thf(mem_c_2Epred__set_2EFINITE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(tp_c_2Epred__set_2EFUNSET,type,
c_2Epred__set_2EFUNSET: del > del > $i ).
thf(mem_c_2Epred__set_2EFUNSET,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC: del > del > $i ).
thf(mem_c_2Epred__set_2EGSPEC,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE: del > del > $i ).
thf(mem_c_2Epred__set_2EIMAGE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EINJ,type,
c_2Epred__set_2EINJ: del > del > $i ).
thf(mem_c_2Epred__set_2EINJ,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT: del > $i ).
thf(mem_c_2Epred__set_2EINSERT,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EINSERT @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER: del > $i ).
thf(mem_c_2Epred__set_2EINTER,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EITSET,type,
c_2Epred__set_2EITSET: del > del > $i ).
thf(mem_c_2Epred__set_2EITSET,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ A_27b ) ) ) ) ).
thf(tp_c_2Epred__set_2ELINV,type,
c_2Epred__set_2ELINV: del > del > $i ).
thf(mem_c_2Epred__set_2ELINV,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ A_27a ) ) ) ) ).
thf(tp_c_2Epred__set_2ELINV__OPT,type,
c_2Epred__set_2ELINV__OPT: del > del > $i ).
thf(mem_c_2Epred__set_2ELINV__OPT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ) ).
thf(tp_c_2Epred__set_2EMAX__SET,type,
c_2Epred__set_2EMAX__SET: $i ).
thf(mem_c_2Epred__set_2EMAX__SET,axiom,
mem @ c_2Epred__set_2EMAX__SET @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ty_2Enum_2Enum ) ).
thf(tp_c_2Epred__set_2EMIN__SET,type,
c_2Epred__set_2EMIN__SET: $i ).
thf(mem_c_2Epred__set_2EMIN__SET,axiom,
mem @ c_2Epred__set_2EMIN__SET @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ty_2Enum_2Enum ) ).
thf(tp_c_2Epred__set_2EPOW,type,
c_2Epred__set_2EPOW: del > $i ).
thf(mem_c_2Epred__set_2EPOW,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Epred__set_2EPREIMAGE,type,
c_2Epred__set_2EPREIMAGE: del > del > $i ).
thf(mem_c_2Epred__set_2EPREIMAGE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EPROD__IMAGE,type,
c_2Epred__set_2EPROD__IMAGE: del > $i ).
thf(mem_c_2Epred__set_2EPROD__IMAGE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EPROD__IMAGE @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Epred__set_2EPROD__SET,type,
c_2Epred__set_2EPROD__SET: $i ).
thf(mem_c_2Epred__set_2EPROD__SET,axiom,
mem @ c_2Epred__set_2EPROD__SET @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ty_2Enum_2Enum ) ).
thf(tp_c_2Epred__set_2EPSUBSET,type,
c_2Epred__set_2EPSUBSET: del > $i ).
thf(mem_c_2Epred__set_2EPSUBSET,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Epred__set_2EREL__RESTRICT,type,
c_2Epred__set_2EREL__RESTRICT: del > $i ).
thf(mem_c_2Epred__set_2EREL__RESTRICT,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Epred__set_2EREST,type,
c_2Epred__set_2EREST: del > $i ).
thf(mem_c_2Epred__set_2EREST,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EREST @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Epred__set_2ERINV,type,
c_2Epred__set_2ERINV: del > del > $i ).
thf(mem_c_2Epred__set_2ERINV,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2ERINV @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ A_27a ) ) ) ) ).
thf(tp_c_2Epred__set_2ESING,type,
c_2Epred__set_2ESING: del > $i ).
thf(mem_c_2Epred__set_2ESING,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ESING @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(tp_c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET: del > $i ).
thf(mem_c_2Epred__set_2ESUBSET,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Epred__set_2ESUM__IMAGE,type,
c_2Epred__set_2ESUM__IMAGE: del > $i ).
thf(mem_c_2Epred__set_2ESUM__IMAGE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ bool ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Epred__set_2ESUM__SET,type,
c_2Epred__set_2ESUM__SET: $i ).
thf(mem_c_2Epred__set_2ESUM__SET,axiom,
mem @ c_2Epred__set_2ESUM__SET @ ( arr @ ( arr @ ty_2Enum_2Enum @ bool ) @ ty_2Enum_2Enum ) ).
thf(tp_c_2Epred__set_2ESURJ,type,
c_2Epred__set_2ESURJ: del > del > $i ).
thf(mem_c_2Epred__set_2ESURJ,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION: del > $i ).
thf(mem_c_2Epred__set_2EUNION,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV: del > $i ).
thf(mem_c_2Epred__set_2EUNIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EUNIV @ A_27a ) @ ( arr @ A_27a @ bool ) ) ).
thf(tp_c_2Epred__set_2Echooser,type,
c_2Epred__set_2Echooser: del > $i ).
thf(mem_c_2Epred__set_2Echooser,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2Echooser @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ).
thf(tp_c_2Epred__set_2Ecount,type,
c_2Epred__set_2Ecount: $i ).
thf(mem_c_2Epred__set_2Ecount,axiom,
mem @ c_2Epred__set_2Ecount @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ).
thf(tp_c_2Epred__set_2Ecountable,type,
c_2Epred__set_2Ecountable: del > $i ).
thf(mem_c_2Epred__set_2Ecountable,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(tp_c_2Epred__set_2Eenumerate,type,
c_2Epred__set_2Eenumerate: del > $i ).
thf(mem_c_2Epred__set_2Eenumerate,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2Eenumerate @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ).
thf(tp_c_2Epred__set_2Eequiv__on,type,
c_2Epred__set_2Eequiv__on: del > $i ).
thf(mem_c_2Epred__set_2Eequiv__on,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2Eequiv__on @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Epred__set_2Eis__measure__maximal,type,
c_2Epred__set_2Eis__measure__maximal: del > $i ).
thf(mem_c_2Epred__set_2Eis__measure__maximal,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2Eis__measure__maximal @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2Enum__to__pair,type,
c_2Epred__set_2Enum__to__pair: $i ).
thf(mem_c_2Epred__set_2Enum__to__pair,axiom,
mem @ c_2Epred__set_2Enum__to__pair @ ( arr @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Epred__set_2Epair__to__num,type,
c_2Epred__set_2Epair__to__num: $i ).
thf(mem_c_2Epred__set_2Epair__to__num,axiom,
mem @ c_2Epred__set_2Epair__to__num @ ( arr @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ty_2Enum_2Enum ) ).
thf(stp_fo_c_2Epred__set_2Epair__to__num,type,
fo__c_2Epred__set_2Epair__to__num: tp__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).
thf(stp_eq_fo_c_2Epred__set_2Epair__to__num,axiom,
! [X0: tp__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum] :
( ( inj__ty_2Enum_2Enum @ ( fo__c_2Epred__set_2Epair__to__num @ X0 ) )
= ( ap @ c_2Epred__set_2Epair__to__num @ ( inj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ X0 ) ) ) ).
thf(tp_c_2Epred__set_2Epairwise,type,
c_2Epred__set_2Epairwise: del > $i ).
thf(mem_c_2Epred__set_2Epairwise,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2Epairwise @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Epred__set_2Epartition,type,
c_2Epred__set_2Epartition: del > $i ).
thf(mem_c_2Epred__set_2Epartition,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2Epartition @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2Eschroeder__close,type,
c_2Epred__set_2Eschroeder__close: del > $i ).
thf(mem_c_2Epred__set_2Eschroeder__close,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2Eschroeder__close @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESPECIFICATION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0P ) )
<=> ( p @ ( ap @ V0P @ V1x ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__APP,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1P ) )
<=> ( p @ ( ap @ V1P @ V0x ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__ABS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V1P @ V2x ) ) ) )
<=> ( p @ ( ap @ V1P @ V0x ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEXTENSION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( V0s = V1t )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ENOT__EQUAL__SETS,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( V0s != V1t )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ENUM__SET__WOP,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ? [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0s ) )
<=> ? [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ V0s ) )
& ! [V3m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EGSPECIFICATION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1v ) @ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b ) @ V0f ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1v ) @ c_2Ebool_2ET )
= ( ap @ V0f @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EGSPECIFICATION__applied,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b ) @ V0f ) @ V1v ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1v ) @ c_2Ebool_2ET )
= ( ap @ V0f @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__ETA,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1x ) @ ( ap @ V0P @ V1x ) ) ) )
= V0P ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__PAIR__ETA,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27b
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) @ ( ap @ ( ap @ V0P @ V1x ) @ V2y ) ) ) ) ) )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) @ V0P ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__GSPEC__IFF,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0y )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ V1P @ V2x ) ) ) ) ) )
<=> ( p @ ( ap @ V1P @ V0y ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__IFF,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V3x ) @ V4y ) ) @ ( ap @ ( ap @ V2P @ V3x ) @ V4y ) ) ) ) ) ) ) )
<=> ( p @ ( ap @ ( ap @ V2P @ V0x ) @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__GSPEC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V3P @ V1y ) )
& ( V2x
= ( ap @ V0f @ V1y ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2x )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ V0f @ V4x ) ) @ ( ap @ V3P @ V4x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__1,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0a ) @ V1b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V4y ) @ V2x ) ) @ ( ap @ V3P @ V4y ) ) ) ) ) )
<=> ( ( p @ ( ap @ V3P @ V0a ) )
& ( V1b = V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__2,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27b @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0a ) @ V1b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b )
@ ( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V4y ) ) @ ( ap @ V3P @ V4y ) ) ) ) ) )
<=> ( ( p @ ( ap @ V3P @ V1b ) )
& ( V0a = V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__same,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27a )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0a ) @ V1b ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V3x ) ) @ ( ap @ V2P @ V3x ) ) ) ) ) )
<=> ( ( p @ ( ap @ V2P @ V0a ) )
& ( V0a = V1b ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESET__MINIMUM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V1M @ V3x ) ) @ ( ap @ V1M @ V4y ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EEMPTY__DEF,axiom,
! [A_27a: del] :
( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( lam @ A_27a
@ ^ [V0x: $i] : c_2Ebool_2EF ) ) ).
thf(conj_thm_2Epred__set_2ENOT__IN__EMPTY,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EMEMBER__NOT__EMPTY,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) )
<=> ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEMPTY__applied,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0x ) )
<=> $false ) ) ).
thf(ax_thm_2Epred__set_2EUNIV__DEF,axiom,
! [A_27a: del] :
( ( c_2Epred__set_2EUNIV @ A_27a )
= ( lam @ A_27a
@ ^ [V0x: $i] : c_2Ebool_2ET ) ) ).
thf(conj_thm_2Epred__set_2EIN__UNIV,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNIV__applied,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( p @ ( ap @ ( c_2Epred__set_2EUNIV @ A_27a ) @ V0x ) ) ) ).
thf(conj_thm_2Epred__set_2EUNIV__NOT__EMPTY,axiom,
! [A_27a: del] :
( ( c_2Epred__set_2EUNIV @ A_27a )
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(conj_thm_2Epred__set_2EEMPTY__NOT__UNIV,axiom,
! [A_27a: del] :
( ( c_2Epred__set_2EEMPTY @ A_27a )
!= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(conj_thm_2Epred__set_2EEQ__UNIV,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) )
<=> ( V0s
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__EQ__UNIV__IMP,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( V0s
= ( c_2Epred__set_2EUNIV @ A_27a ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1v ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ESUBSET__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__THM,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0P ) @ V1Q ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__applied,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0s @ V2x ) )
=> ( p @ ( ap @ V1t @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__TRANS,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__REFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__ANTISYM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) ) )
=> ( V0s = V1t ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEMPTY__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0s ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__EMPTY,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__UNIV,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNIV__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0s ) )
<=> ( V0s
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEQ__SUBSET__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( V0s = V1t )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__SUBSET__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) ) )
<=> ( V0s = V1t ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__ADD,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2d: tp__ty_2Enum_2Enum] :
( ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) @ ( ap @ V0f @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V2d ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EK__SUBSET,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ bool @ A_27a ) @ V0x ) ) @ V1y ) )
<=> ( ~ ( p @ V0x )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__K,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ bool )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0x ) @ ( ap @ ( c_2Ecombin_2EK @ bool @ A_27a ) @ V1y ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
| ( p @ V1y ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EPSUBSET__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ V1t ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ( V0s != V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__TRANS,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__IRREFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ V0s ) ) ) ).
thf(conj_thm_2Epred__set_2ENOT__PSUBSET__EMPTY,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2ENOT__UNIV__PSUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0s ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__UNIV,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EUNION__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__applied,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) @ V2x ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__ASSOC,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1t ) @ V2u ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ V2u ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__IDEMPOT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V0s )
= V0s ) ) ).
thf(conj_thm_2Epred__set_2EUNION__COMM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1t ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__UNION,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V3t ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ V2u ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V2u ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__UNION__ABSORPTION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t )
= V1t ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__EMPTY,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0s )
= V0s ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V1s ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__UNIV,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0s )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEMPTY__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( V1t
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFORALL__IN__UNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( p @ ( ap @ V0P @ V3x ) ) ) )
<=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) )
=> ( p @ ( ap @ V0P @ V4x ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2t ) )
=> ( p @ ( ap @ V0P @ V5x ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EINTER__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__INTER,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__applied,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) @ V2x ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__ASSOC,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1t ) @ V2u ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) @ V2u ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__IDEMPOT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V0s )
= V0s ) ) ).
thf(conj_thm_2Epred__set_2EINTER__COMM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1t ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__SUBSET,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) @ V0s ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V3t ) @ V2s ) ) @ V2s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__INTER,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1t ) @ V2u ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V2u ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__INTER__ABSORPTION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t )
= V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__INTER1,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t )
= V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__INTER2,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1t ) @ V0s )
= V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__EMPTY,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__UNIV,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0s )
= V0s ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V1s ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__OVER__INTER,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1t ) @ V2u ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__OVER__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1t ) @ V2u ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EDISJOINT__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V1t ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__DISJOINT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V1t ) )
<=> ~ ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__SYM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__ALT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V1t ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__EMPTY,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__EMPTY__REFL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__EMPTY__REFL__RWT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V0s ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ V2u ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1t ) @ V2u ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__UNION__BOTH,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) @ V2u ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1t ) @ V2u ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V2u ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1t ) @ V2u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2u ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EDIFF__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2x ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__DIFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__applied,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) @ V2x ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__EMPTY,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V0s ) ) ).
thf(conj_thm_2Epred__set_2EEMPTY__DIFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__UNIV,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__DIFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) @ V1t )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__EQ__EMPTY,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__DIFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1t ) @ V0s ) )
= V1t )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1t ) @ V0s ) ) @ V0s )
= V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__DIFF__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) )
= V1t ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__UNION,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ A_27a @ bool ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1y ) @ V2z ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0x ) @ V1y ) ) @ V2z ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__COMM,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ A_27a @ bool ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0x ) @ V1y ) ) @ V2z )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0x ) @ V2z ) ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__SAME__UNION,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0x ) @ V1y ) ) @ V0x )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1y ) @ V0x ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0x ) @ V1y ) ) @ V1y )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0x ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__INTER,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) @ V2g )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V2g ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__INTER2,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1t ) @ V0s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__DIFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1t ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__DIFFS,axiom,
! [A_27a: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( ( ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V1f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) )
& ! [V5n: tp__ty_2Enum_2Enum] :
( ( ap @ V0g @ ( inj__ty_2Enum_2Enum @ V5n ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ V1f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V5n ) ) ) )
& ( V2m != V3n ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V0g @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( ap @ V0g @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EINSERT__DEF,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2y ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2y ) @ V0x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1y ) @ V2s ) ) )
<=> ( ( V0x = V1y )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__applied,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1y ) @ V2s ) @ V0x ) )
<=> ( ( V0x = V1y )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOMPONENT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESET__CASES,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V2t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDECOMPOSITION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) )
<=> ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V2t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EABSORPTION,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s )
= V1s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EABSORPTION__RWT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s )
= V1s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__COMM,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1y ) @ V2s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1y ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__UNIV,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2ENOT__INSERT__EMPTY,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s )
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2ENOT__EMPTY__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( c_2Epred__set_2EEMPTY @ A_27a )
!= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__UNION,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) @ V2t )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__UNION__EQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) @ V2t )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__INTER,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) @ V2t )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) @ V2t ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1s ) @ V2t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__INSERT_27,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V2t ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V2t ) @ V1s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__SUBSET,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) @ V2t ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V2t ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__DIFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V0s ) ) @ V1t )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNIV__BOOL,axiom,
( ( c_2Epred__set_2EUNIV @ bool )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ bool ) @ c_2Ebool_2ET ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ bool ) @ c_2Ebool_2EF ) @ ( c_2Epred__set_2EEMPTY @ bool ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFORALL__IN__INSERT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ V2s ) ) )
=> ( p @ ( ap @ V0P @ V3x ) ) ) )
<=> ( ( p @ ( ap @ V0P @ V1a ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
=> ( p @ ( ap @ V0P @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEXISTS__IN__INSERT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ V2s ) ) )
& ( p @ ( ap @ V0P @ V3x ) ) )
<=> ( ( p @ ( ap @ V0P @ V1a ) )
| ? [V4x: $i] :
( ( mem @ V4x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
& ( p @ ( ap @ V0P @ V4x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EDELETE__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V1x )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__DELETE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V2y ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) )
& ( V1x != V2y ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__applied,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V2y ) @ V1x ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) )
& ( V1x != V2y ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__NON__ELEMENT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x )
= V1s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__NON__ELEMENT__RWT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V1x )
= V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__DELETE__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2x_27: $i] :
( ( mem @ V2x_27 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x_27 ) @ V0s ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V2x_27 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x_27 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V1x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEMPTY__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EELT__IN__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x ) ) @ V0x )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__COMM,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V0x ) ) @ V1y )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1y ) ) @ V0x ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__SUBSET,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x ) ) @ V1s ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2t ) @ V0x ) ) )
<=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__INSERT__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V2t ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x ) ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__OF__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__INSERT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V2x ) ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__INSERT__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ V1t ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V0s ) ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__MEMBER,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ V1t ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
& ? [V2y: $i] :
( ( mem @ V2y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V1t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V2s ) ) @ V1y )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1y ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1y ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x ) )
= V1s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__INTER,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V2x ) ) @ V1t )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) @ V2x ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__DELETE__SYM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V2x ) ) @ V1t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1t ) @ V2x ) ) @ V0s ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ECHOICE__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) @ V0s ) ) ) ) ).
thf(ax_thm_2Epred__set_2EREST__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__REST,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V1s ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
& ( V0x
!= ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EREST__applied,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V1s ) @ V0x ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
& ( V0x
!= ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECHOICE__NOT__IN__REST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECHOICE__INSERT__REST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) )
= V0s ) ) ) ).
thf(conj_thm_2Epred__set_2EREST__SUBSET,conjecture,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) @ V0s ) ) ) ).
%------------------------------------------------------------------------------