ITP001 Axioms: ITP027^5.ax
%------------------------------------------------------------------------------
% File : ITP027^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, 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 : pred_set^2.ax [Gau20]
% : HL4027^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 687 ( 34 unt; 51 typ; 0 def)
% Number of atoms : 10946 ( 432 equ; 0 cnn)
% Maximal formula atoms : 58 ( 15 avg)
% Number of connectives : 20635 ( 106 ~; 28 |; 336 &;18332 @)
% ( 197 <=>;1636 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 13 avg;18332 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 57 ( 57 >; 0 *; 0 +; 0 <<)
% Number of symbols : 128 ( 126 usr; 83 con; 0-2 aty)
% Number of variables : 2276 ( 68 ^2129 !; 79 ?;2276 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
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,axiom,
! [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 ) ) ) ).
thf(conj_thm_2Epred__set_2EREST__PSUBSET,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_2EPSUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) @ V0s ) ) ) ) ).
thf(ax_thm_2Epred__set_2ESING__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V0s ) )
<=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESING__EMPTY,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> $false ) ).
thf(conj_thm_2Epred__set_2ESING__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) )
<=> ( ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V1s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESING__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) )
<=> ( ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V0s ) )
& ( V1t
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
| ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V1t ) )
& ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
| ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V1t ) )
& ( V0s = V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1y ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
<=> ( V0x = V1y ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESING__applied,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1y ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0x ) )
<=> ( V0x = V1y ) ) ) ) ).
thf(conj_thm_2Epred__set_2ENOT__SING__EMPTY,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2ENOT__EMPTY__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( c_2Epred__set_2EEMPTY @ A_27a )
!= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEQUAL__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1y ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( V0x = V1y ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__SING__EMPTY,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__SING__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESING__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V0x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__EQ__SING,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 )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECHOICE__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= V0x ) ) ).
thf(conj_thm_2Epred__set_2EREST__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2ESING__IFF__EMPTY__REST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V0s ) )
<=> ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EIMAGE__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( ( mem @ V0y @ A_27b )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V0y ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V1s ) ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( V0y
= ( ap @ V2f @ V3x ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__applied,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( ( mem @ V0y @ A_27b )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V1s ) @ V0y ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( V0y
= ( ap @ V2f @ V3x ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__IN,axiom,
! [A_27a: del,A_27b: 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 ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V0x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__ID,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : V1x ) )
@ V0s )
= V0s ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__I,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ V0s )
= V0s ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__II,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ bool ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__COMPOSE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V0f ) @ V1g ) ) @ V2s )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1g ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__INSERT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V2s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ ( ap @ V0f @ V1x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__EQ__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27b ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__DELETE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2s ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1x ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__UNION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__SUBSET,axiom,
! [A_27a: del,A_27b: 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 ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V0s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__INTER,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__11,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) )
<=> ( V3x = V4y ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s1 )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s2 ) )
<=> ( V1s1 = V2s2 ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( ap @ V0f @ V3x )
= ( ap @ V0f @ V4y ) )
<=> ( V3x = V4y ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s2 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1s1 ) @ V2s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__CONG,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f_27: $i] :
( ( mem @ V2f_27 @ ( arr @ A_27a @ A_27b ) )
=> ! [V3s_27: $i] :
( ( mem @ V3s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( V1s = V3s_27 )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3s_27 ) )
=> ( ( ap @ V0f @ V4x )
= ( ap @ V2f_27 @ V4x ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f_27 ) @ V3s_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27b ) @ V0f )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ A_27a @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) @ ( c_2Epair_2EFST @ A_27a @ bool ) ) @ V0f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ bool @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) @ ( c_2Epair_2ESND @ A_27a @ bool ) ) @ V0f ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__IMAGE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27c @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27c @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27c @ A_27a ) @ V1g ) @ V2s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a ) @ V0f ) @ V1g ) ) @ V2s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFORALL__IN__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V1f ) @ V2s ) ) )
=> ( p @ ( ap @ V0P @ V3y ) ) ) )
<=> ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4x ) @ V2s ) )
=> ( p @ ( ap @ V0P @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEXISTS__IN__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V1f ) @ V2s ) ) )
& ( p @ ( ap @ V0P @ V3y ) ) )
<=> ? [V4x: $i] :
( ( mem @ V4x @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4x ) @ V2s ) )
& ( p @ ( ap @ V0P @ ( ap @ V1f @ V4x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EINJ__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V1s ) ) )
=> ( ( ( ap @ V0f @ V4x )
= ( ap @ V0f @ V5y ) )
=> ( V4x = V5y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__IFF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V1s ) ) )
=> ( ( ( ap @ V0f @ V4x )
= ( ap @ V0f @ V5y ) )
<=> ( V4x = V5y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__ID,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : V1x ) )
@ V0s )
@ V0s ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__COMPOSE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27b @ A_27c ) @ V1g ) @ V3t ) @ V4u ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1g ) @ V0f ) ) @ V2s ) @ V4u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1s ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
<=> ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__DELETE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V2f ) @ V0s ) @ V1t ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3e ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V3e ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27b ) @ V1t ) @ ( ap @ V2f @ V3e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__INSERT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V2s ) ) @ V3t ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V1x ) ) @ V3t ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s ) )
& ( ( ap @ V0f @ V1x )
= ( ap @ V0f @ V4y ) ) )
=> ( V1x = V4y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__EXTEND,axiom,
! [A_27a: del,A_27b: del,V0b: $i] :
( ( mem @ V0b @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0b ) @ V1s ) @ V2t ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EUPDATE @ A_27a @ A_27b ) @ V3x ) @ V4y ) @ V0b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V4y ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ! [V3s0: $i] :
( ( mem @ V3s0 @ ( arr @ A_27a @ bool ) )
=> ! [V4t0: $i] :
( ( mem @ V4t0 @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3s0 ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V2t ) @ V4t0 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V3s0 ) @ V4t0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__IMAGE__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ V2t ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ESURJ__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4x ) @ V2t ) )
=> ? [V5y: $i] :
( ( mem @ V5y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V1s ) )
& ( ( ap @ V0f @ V5y )
= V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESURJ__ID,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : V1x ) )
@ V0s )
@ V0s ) ) ) ).
thf(conj_thm_2Epred__set_2ESURJ__COMPOSE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27b @ A_27c ) @ V1g ) @ V3t ) @ V4u ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1g ) @ V0f ) ) @ V2s ) @ V4u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESURJ__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1s ) )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
<=> ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__SURJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
<=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s )
= V2t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESURJ__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESURJ__IMP__INJ,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V2f ) @ V0s ) @ V1t ) ) )
=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27b @ A_27a ) @ V3g ) @ V1t ) @ V0s ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EBIJ__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__ID,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : V1x ) )
@ V0s )
@ V0s ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__IMP__11,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ V0f @ V1x )
= ( ap @ V0f @ V2y ) )
<=> ( V1x = V2y ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1s ) )
<=> ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
<=> ( V2s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__COMPOSE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ! [V4u: $i] :
( ( mem @ V4u @ ( arr @ A_27c @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27b @ A_27c ) @ V1g ) @ V3t ) @ V4u ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1g ) @ V0f ) ) @ V2s ) @ V4u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__DELETE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V2f ) @ V0s ) @ V1t ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3e ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V3e ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27b ) @ V1t ) @ ( ap @ V2f @ V3e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__IMAGE__BIJ,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ? [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V1f ) @ V0s ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V1f ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__BIJ__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0s0: $i] :
( ( mem @ V0s0 @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s0 ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V2f ) @ V1s ) @ V3t ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V2f ) @ V0s0 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V2f ) @ V0s0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__SYM__IMP,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V2f ) @ V0s ) @ V1t ) ) )
=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27b @ A_27a ) @ V3g ) @ V1t ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__SYM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V2f ) @ V0s ) @ V1t ) ) )
<=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27b @ A_27a ) @ V3g ) @ V1t ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27c @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27b @ bool ) )
=> ( ( ? [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27c ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27c ) @ V3f ) @ V0s ) @ V1t ) ) )
& ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27c @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27c @ A_27b ) @ V4g ) @ V1t ) @ V2u ) ) ) )
=> ? [V5h: $i] :
( ( mem @ V5h @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V5h ) @ V0s ) @ V2u ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__INV,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27b @ A_27a ) @ V3g ) @ V2t ) @ V1s ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27b ) @ V3g ) @ V0f ) @ V4x )
= V4x ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5x ) @ V2t ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ A_27b @ A_27a ) @ V0f ) @ V3g ) @ V5x )
= V5x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EFUNSET,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V0P ) @ V1Q )
= ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V2f: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ V1Q ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EDFUNSET,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b ) @ V0P ) @ V1Q )
= ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V2f: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ ( ap @ V1Q @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__FUNSET,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V1P ) @ V2Q ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2Q ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFUNSET__applied,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V1P ) @ V2Q ) @ V0f ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2Q ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__DFUNSET,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b ) @ V1P ) @ V2Q ) ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ ( ap @ V2Q @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDFUNSET__applied,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b ) @ V1P ) @ V2Q ) @ V0f ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ ( ap @ V2Q @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFUNSET__THM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNIV__FUNSET__UNIV,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
= ( c_2Epred__set_2EUNIV @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Epred__set_2EFUNSET__DFUNSET,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V0x ) @ V1y )
= ( ap @ ( ap @ ( c_2Epred__set_2EDFUNSET @ A_27a @ A_27b ) @ V0x ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27b @ bool ) @ A_27a ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEMPTY__FUNSET,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0s )
= ( c_2Epred__set_2EUNIV @ ( arr @ A_27a @ A_27b ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFUNSET__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFUNSET__INTER,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ bool ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V0a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V1b ) @ V2c ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ( arr @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V0a ) @ V1b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V0a ) @ V2c ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Eschroeder__close__def,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2Eschroeder__close @ A_27a ) @ V0f ) @ V1s ) @ V2x ) )
<=> ? [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V0f ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESCHROEDER__CLOSE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2Eschroeder__close @ A_27a ) @ V1f ) @ V2s ) ) )
<=> ? [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESCHROEDER__CLOSED,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2Eschroeder__close @ A_27a ) @ V0f ) @ V1s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2Eschroeder__close @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESCHROEDER__CLOSE__SUBSET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ 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_2Eschroeder__close @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESCHROEDER__CLOSE__SET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27a ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27a ) @ V1s ) @ V1s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2Eschroeder__close @ A_27a ) @ V0f ) @ V2t ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESCHROEDER__BERNSTEIN__AUTO,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 ) )
& ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27a ) @ V2f ) @ V0s ) @ V1t ) ) ) )
=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27a ) @ V3g ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESCHROEDER__BERNSTEIN,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V2f ) @ V0s ) @ V1t ) ) )
& ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27b @ A_27a ) @ V3g ) @ V1t ) @ V0s ) ) ) )
=> ? [V4h: $i] :
( ( mem @ V4h @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V4h ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__INJ__SURJ,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V2f ) @ V0s ) @ V1t ) ) )
& ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V3g ) @ V0s ) @ V1t ) ) ) )
=> ? [V4h: $i] :
( ( mem @ V4h @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V4h ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__ALT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ A_27b ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EFUNSET @ A_27a @ A_27b ) @ V1s ) @ V2t ) ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V2t ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V3y ) @ ( ap @ V0f @ V4x ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__INSERT__IMP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1e ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s ) ) @ V3t ) ) )
=> ? [V4u: $i] :
( ( mem @ V4u @ ( arr @ A_27b @ bool ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ ( ap @ V0f @ V1e ) ) @ V4u )
= V3t )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V1e ) ) @ V4u ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ V4u ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ( V2t
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ELINV__OPT__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2y )
= ( ap
@ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2y ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
@ ( ap @ ( c_2Eoption_2ESOME @ A_27a )
@ ( ap @ ( c_2Emin_2E_40 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2y ) ) ) ) ) )
@ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ELINV__OPT__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2y )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3x ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
& ( ( ap @ V0f @ V3x )
= V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__LINV__OPT__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV__OPT @ A_27b @ A_27a ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( c_2Eoption_2ESOME @ A_27b ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__LINV__OPT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V4y )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3x ) )
<=> ( ( V4y
= ( ap @ V0f @ V3x ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V2t ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ELINV__LO,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2y )
= ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ELINV__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ ( ap @ V0f @ V3x ) )
= V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__LINV__INV,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ V2t ) )
=> ( ( ap @ V0f @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V3x ) )
= V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__LINV__BIJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ V2t ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__IFF__INV,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2t ) ) ) )
& ? [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27b @ A_27a ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V5x ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V4g @ V5x ) ) @ V1s ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6x ) @ V1s ) )
=> ( ( ap @ V4g @ ( ap @ V0f @ V6x ) )
= V6x ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7x ) @ V2t ) )
=> ( ( ap @ V0f @ ( ap @ V4g @ V7x ) )
= V7x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__INSERT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s ) ) @ V3t ) )
<=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1e ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V1e ) ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27b ) @ V3t ) @ ( ap @ V0f @ V1e ) ) ) ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1e ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ V3t ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ERINV__LO,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ERINV @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2y )
= ( ap @ ( c_2Eoption_2ETHE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ELINV__OPT @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ERINV__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ V2t ) )
=> ( ( ap @ V0f @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ERINV @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V3x ) )
= V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESURJ__INJ__INV,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ? [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27b @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27b @ A_27a ) @ V3g ) @ V2t ) @ V1s ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4y ) @ V2t ) )
=> ( ( ap @ V0f @ ( ap @ V3g @ V4y ) )
= V4y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EFINITE__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ V1P @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V1P @ V2s ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( p @ ( ap @ V1P @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3e ) @ V2s ) ) ) ) ) ) )
=> ( p @ ( ap @ V1P @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__EMPTY,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ V0P @ V1s ) ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ V1s ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2e ) @ V1s ) ) ) ) ) ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s ) )
=> ( p @ ( ap @ V0P @ V3s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__DELETE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V0x ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__REST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__REST__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__FINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__FINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__FINITE__I,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__FINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__DIFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__DIFF__down,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0P ) @ V1Q ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1Q ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESING__FINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__FINITE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITELY__INJECTIVE__IMAGE__FINITE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( 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_2Emin_2E_3D @ A_27b ) @ V1x ) @ ( ap @ V0f @ V2y ) ) ) ) ) ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V3s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJECTIVE__IMAGE__FINITE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ V0f @ V1x )
= ( ap @ V0f @ V2y ) )
<=> ( V1x = V2y ) ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V3s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__INJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EREL__RESTRICT__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V0R ) @ V1s ) @ V2x ) @ V3y ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EREL__RESTRICT__EMPTY,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V0R ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EREL__RESTRICT__SUBSET,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s1 ) @ V1s2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V2R ) @ V0s1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V2R ) @ V1s2 ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ECARD__DEF,axiom,
! [A_27a: del] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= fo__c_2Enum_2E0 )
& ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__EMPTY,axiom,
! [A_27a: del] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Epred__set_2ECARD__INSERT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__EQ__0,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= fo__c_2Enum_2E0 )
<=> ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__DELETE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V1x ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__INTER__LESS__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__UNION__EQN,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1t ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1t ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__PSUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1t ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__EQ__CARD,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) ) )
=> ( V0s = V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESING__IFF__CARD1,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2ESING @ A_27a ) @ V0s ) )
<=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__DIFF,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0t ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V0t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V0t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__DIFF__EQN,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V0t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V0t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ELESS__CARD__DIFF,axiom,
! [A_27a: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0t ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0t ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V0t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__FINITE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__FINITE__SUBSET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ A_27a ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) )
=> ? [V3N: tp__ty_2Enum_2Enum] :
! [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3N ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__BIJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__BIJ__CARD,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__BIJ__CARD__EQ,axiom,
! [A_27a: del,A_27b: del,V0S: $i] :
( ( mem @ V0S @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0S ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V2f ) @ V0S ) @ V1t ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0S ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__INJ__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( ap @ V0f @ V2x )
= ( ap @ V0f @ V3y ) )
<=> ( V2x = V3y ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESURJ__CARD,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V2t ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__SURJ,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V1f ) @ V0s ) @ V2t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__SURJ__BIJ,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27b ) @ V1f ) @ V0s ) @ V2t ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V2t ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V1f ) @ V0s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__COMPLETE__INDUCTION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ( ! [V2y: $i] :
( ( mem @ V2y @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V2y ) @ V1x ) )
=> ( p @ ( ap @ V0P @ V2y ) ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1x ) )
=> ( p @ ( ap @ V0P @ V1x ) ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3x ) )
=> ( p @ ( ap @ V0P @ V3x ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__CARD__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V2s ) @ V1t ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__CARD,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPHP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V2t ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) )
=> ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__CARD__IMAGE__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Ecount__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ty_2Enum_2Enum @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1m: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ bool ) @ V1m ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V1m ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__COUNT,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNT__applied,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
<=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNT__ZERO,axiom,
( ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Epred__set_2ECOUNT__SUC,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ c_2Epred__set_2Ecount @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__COUNT,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__COUNT,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) )
= V0n ) ).
thf(conj_thm_2Epred__set_2ECOUNT__11,axiom,
! [V0n1: tp__ty_2Enum_2Enum,V1n2: tp__ty_2Enum_2Enum] :
( ( ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n1 ) )
= ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1n2 ) ) )
<=> ( V0n1 = V1n2 ) ) ).
thf(conj_thm_2Epred__set_2ENOT__IN__FINITE,axiom,
! [A_27a: del] :
( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINFINITE__INHAB,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0P ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__11__INFINITE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ V0f @ V1x )
= ( ap @ V0f @ V2y ) )
=> ( V1x = V2y ) ) ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V3s ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINFINITE__SUBSET,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__INFINITE__NOT__FINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ 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_2EINFINITE__INJ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
& ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINFINITE__INJ__NOT__SURJ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27a ) @ V1f ) @ V0s ) @ V0s ) )
& ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ A_27a ) @ V1f ) @ V0s ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINFINITE__UNIV,axiom,
! [A_27a: del] :
( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ? [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ V0f @ V1x )
= ( ap @ V0f @ V2y ) )
=> ( V1x = V2y ) ) ) )
& ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ap @ V0f @ V4x )
!= V3y ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINFINITE__NUM__UNIV,axiom,
~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__PSUBSET__INFINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__PSUBSET__UNIV,axiom,
! [A_27a: del] :
( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
<=> ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINFINITE__DIFF__FINITE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t )
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__BIJ__COUNT__EQ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ? [V1c: $i] :
( ( mem @ V1c @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ? [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ A_27a ) @ V1c ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__BIJ__COUNT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ? [V2b: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V2b ) ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__ISO__NUM,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V2n: tp__ty_2Enum_2Enum,V3m: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) )
=> ( ( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) )
= ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3m ) ) )
=> ( V2n = V3m ) ) )
& ( V0s
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ ( ap @ V1f @ V4n ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V4n ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__WEAK__ENUMERATE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ? [V2b: tp__ty_2Enum_2Enum] :
! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3e ) @ V0s ) )
<=> ? [V4n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4n ) ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) )
& ( V3e
= ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__WF__noloops,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V0R ) @ V1s ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V0R ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__StrongOrder__WF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Erelation_2EStrongOrder @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V0R ) @ V1s ) ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a ) @ V0R ) @ V1s ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EBIGUNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0P )
= ( 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 @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2s: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__BIGUNION,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1sos: $i] :
( ( mem @ V1sos @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1sos ) ) )
<=> ? [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ V1sos ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__applied,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1sos: $i] :
( ( mem @ V1sos @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1sos ) @ V0x ) )
<=> ? [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ V1sos ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__BIGUNION__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2y ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( arr @ A_27b @ bool ) ) @ V0f ) @ V1s ) ) ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2y ) @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__EQ__EMPTY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0P )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( ( V0P
= ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) )
| ( V0P
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) ) ) )
& ( ( ( c_2Epred__set_2EEMPTY @ A_27a )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0P ) )
<=> ( ( V0P
= ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) )
| ( V0P
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) )
= V0x ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__PAIR,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V1t ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__UNION,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( arr @ A_27a @ bool ) ) @ V0s1 ) @ V1s2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s1 ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1s2 ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__BIGUNION,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s ) ) @ V1t ) )
<=> ! [V2s_27: $i] :
( ( mem @ V2s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s_27 ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V2s_27 ) @ V1t ) ) ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V4t ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V3s ) ) )
<=> ! [V5s_27: $i] :
( ( mem @ V5s_27 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V5s_27 ) @ V3s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V4t ) @ V5s_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__INSERT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V0s ) @ V1P ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1P ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__SUBSET,axiom,
! [A_27a: del,V0X: $i] :
( ( mem @ V0X @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1P ) ) @ V0X ) )
<=> ! [V2Y: $i] :
( ( mem @ V2Y @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2Y ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2Y ) @ V0X ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__IMAGE__UNIV,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ! [V1N: tp__ty_2Enum_2Enum] :
( ! [V2n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1N ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
=> ( ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V0f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1N ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__BIGUNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V0P ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ V0P ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0P ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__BIGUNION__EQ,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0P ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V0P ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ V0P ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__BIGUNION__I,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V0x ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0x ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1P ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__BIGUNION__SAME__SIZED__SETS,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V1s ) )
& ! [V2e: $i] :
( ( mem @ V2e @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2e ) @ V1s ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2e ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V2e ) )
= V0n ) ) ) )
& ! [V3e1: $i] :
( ( mem @ V3e1 @ ( arr @ A_27a @ bool ) )
=> ! [V4e2: $i] :
( ( mem @ V4e2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3e1 ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4e2 ) @ V1s ) )
& ( V3e1 != V4e2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V3e1 ) @ V4e2 ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ ( arr @ A_27a @ bool ) ) @ V1s ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__COUNT,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1m: tp__ty_2Enum_2Enum,V2n: tp__ty_2Enum_2Enum] :
( ( V1m != V2n )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
=> ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( arr @ A_27a @ bool ) ) @ V0f ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFORALL__IN__BIGUNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1s ) ) )
=> ( p @ ( ap @ V0P @ V2x ) ) ) )
<=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3t ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V3t ) ) )
=> ( p @ ( ap @ V0P @ V4x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EBIGINTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V0P )
= ( 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 @ ( c_2Ebool_2E_21 @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2s: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__BIGINTER,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1B ) ) )
<=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2P ) @ V1B ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2P ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGINTER__applied,axiom,
! [A_27a: del,V0B: $i] :
( ( mem @ V0B @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V0B ) @ V1x ) )
<=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2P ) @ V0B ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2P ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__BIGINTER__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( arr @ A_27a @ bool ) ) @ V1f ) @ V2s ) ) ) )
<=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ V1f @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGINTER__INSERT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V0P ) @ V1B ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0P ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1B ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGINTER__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(conj_thm_2Epred__set_2EBIGINTER__INTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V1Q ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0P ) @ V1Q ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGINTER__SING,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V0P ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) )
= V0P ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__BIGINTER,axiom,
! [A_27a: del,V0X: $i] :
( ( mem @ V0X @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0X ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1P ) ) )
<=> ! [V2Y: $i] :
( ( mem @ V2Y @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2Y ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0X ) @ V2Y ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDISJOINT__BIGINTER,axiom,
! [A_27a: del,V0X: $i] :
( ( mem @ V0X @ ( arr @ A_27a @ bool ) )
=> ! [V1Y: $i] :
( ( mem @ V1Y @ ( arr @ A_27a @ bool ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1Y ) @ V2P ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V1Y ) @ V0X ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0X ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V2P ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V2P ) ) @ V0X ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGINTER__UNION,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( arr @ A_27a @ bool ) ) @ V0s1 ) @ V1s2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V0s1 ) ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1s2 ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGINTER__SUBSET,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V0sp ) ) ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1s ) ) @ V0sp ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__BIGINTER1,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0sp ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1s ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2u: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0sp ) @ V2u ) ) )
@ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__BIGINTER,axiom,
! [A_27a: del,V0sp: $i] :
( ( mem @ V0sp @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V0sp ) ) ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0sp )
@ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3u: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0sp ) @ V3u ) ) )
@ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__BIGINTER,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ? [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ V0P ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V0P ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ECROSS__DEF,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V2p: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) @ V2p ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2p ) ) @ V0P ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V2p ) ) @ V1Q ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__CROSS,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V2x ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2x ) ) @ V0P ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V2x ) ) @ V1Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__applied,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q ) @ V2x ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2x ) ) @ V0P ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V2x ) ) @ V1Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__EMPTY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
= ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27c @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27c ) ) @ V0P )
= ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27c @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__EMPTY__EQN,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t )
= ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__INSERT__LEFT,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V0P ) ) @ V1Q )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V1Q ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__INSERT__RIGHT,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V2x ) @ V1Q ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V2x ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__CROSS,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1Q ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__SINGS,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V1y ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__SING__CROSS,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1P ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V1P ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1P ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__CROSS,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1Q ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27b ) @ V1Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2P0: $i] :
( ( mem @ V2P0 @ ( arr @ A_27a @ bool ) )
=> ! [V3Q0: $i] :
( ( mem @ V3Q0 @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V2P0 ) @ V3Q0 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q ) ) )
<=> ( ( V2P0
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V3Q0
= ( c_2Epred__set_2EEMPTY @ A_27b ) )
| ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2P0 ) @ V0P ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V3Q0 ) @ V1Q ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__CROSS__EQ,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0P ) @ V1Q ) ) )
<=> ( ( V0P
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V1Q
= ( c_2Epred__set_2EEMPTY @ A_27b ) )
| ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V1Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__UNIV,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Epred__set_2EUNIV @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
= ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ).
thf(conj_thm_2Epred__set_2EINFINITE__PAIR__UNIV,axiom,
! [A_27a: del,A_27b: del] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__UNIV,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( c_2Esum_2EINL @ A_27a @ A_27b ) ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( c_2Esum_2EINR @ A_27a @ A_27b ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__INL,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2x ) ) @ V1t ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( c_2Esum_2EINL @ A_27a @ A_27b ) ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINJ__INR,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ( ty_2Esum_2Esum @ A_27b @ A_27a ) @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Esum_2Esum @ A_27b @ A_27a ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27b @ A_27a ) @ V2x ) ) @ V1t ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ A_27a ) ) @ ( c_2Esum_2EINR @ A_27b @ A_27a ) ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ECOMPL__DEF,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0P )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ V0P ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__COMPL,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_2ECOMPL @ A_27a ) @ V1s ) ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOMPL__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_2ECOMPL @ A_27a ) @ V1s ) @ V0x ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOMPL__COMPL,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0s ) )
= V0s ) ) ).
thf(conj_thm_2Epred__set_2ECOMPL__CLAUSES,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0s ) ) @ V0s )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0s ) ) @ V0s )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOMPL__SPLITS,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( arr @ A_27a @ bool ) )
=> ! [V1q: $i] :
( ( mem @ V1q @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0p ) @ V1q ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0p ) ) @ V1q ) )
= V1q ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__UNION__COMPL,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_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOMPL__EMPTY,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(conj_thm_2Epred__set_2ECOMPL__INTER,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0x ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0x ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0x ) ) @ V0x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOMPL__UNION,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EITSET__ind,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V2s ) ) @ ( ap @ ( ap @ V0f @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V2s ) ) @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2s ) @ V3b ) ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ ( arr @ A_27a @ bool ) )
=> ! [V5v1: $i] :
( ( mem @ V5v1 @ A_27b )
=> ( p @ ( ap @ ( ap @ V1P @ V4v ) @ V5v1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EITSET__def,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V1f ) @ V0s ) @ V2b )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V2b ) @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) @ ( ap @ ( ap @ V1f @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) @ V2b ) ) ) ) @ ( c_2Ebool_2EARB @ A_27b ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EITSET__IND,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
& ( V2s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V2s ) ) @ ( ap @ ( ap @ V0f @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V2s ) ) @ V3b ) ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2s ) @ V3b ) ) ) ) )
=> ! [V4v: $i] :
( ( mem @ V4v @ ( arr @ A_27a @ bool ) )
=> ! [V5v1: $i] :
( ( mem @ V5v1 @ A_27b )
=> ( p @ ( ap @ ( ap @ V1P @ V4v ) @ V5v1 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EITSET__THM,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V1f ) @ V0s ) @ V2b )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V2b ) @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) @ ( ap @ ( ap @ V1f @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) @ V2b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EITSET__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V1b )
= V1b ) ) ) ).
thf(conj_thm_2Epred__set_2EITSET__INSERT,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V0s ) ) @ V3b )
= ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V0s ) ) ) @ ( ap @ ( ap @ V1f @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V0s ) ) ) @ V3b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOMMUTING__ITSET__INSERT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27b )
=> ( ( ap @ ( ap @ V0f @ V2x ) @ ( ap @ ( ap @ V0f @ V3y ) @ V4z ) )
= ( ap @ ( ap @ V0f @ V3y ) @ ( ap @ ( ap @ V0f @ V2x ) @ V4z ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6b: $i] :
( ( mem @ V6b @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V5x ) @ V1s ) ) @ V6b )
= ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V5x ) ) @ ( ap @ ( ap @ V0f @ V5x ) @ V6b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOMMUTING__ITSET__RECURSES,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27b )
=> ( ( ap @ ( ap @ V0f @ V4x ) @ ( ap @ ( ap @ V0f @ V5y ) @ V6z ) )
= ( ap @ ( ap @ V0f @ V5y ) @ ( ap @ ( ap @ V0f @ V4x ) @ V6z ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s ) ) @ V3b )
= ( ap @ ( ap @ V0f @ V1e ) @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1e ) ) @ V3b ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ESUM__IMAGE__DEF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap
@ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ ty_2Enum_2Enum )
@ ( lam @ A_27a
@ ^ [V2e: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V3acc: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ V2e ) ) @ V3acc ) ) ) )
@ V1s )
@ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__THM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= fo__c_2Enum_2E0 )
& ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ V0f @ V1e ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1e ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__SING,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V1e ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__SUBSET__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__IN__LE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V2e ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__DELETE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s ) @ V2e ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ V1s ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) @ ( ap @ V0f @ V2e ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__UNION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V2t ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__lower__bound,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ V0f @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__upper__bound,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V3x ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__SAME__IMAGE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0P ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2p ) @ V0P ) )
& ! [V3q: $i] :
( ( mem @ V3q @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3q ) @ V0P ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V1f @ V2p ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V1f @ V3q ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V1f ) @ V0P ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0P ) ) @ ( ap @ V1f @ V2p ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__CONG,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( V0s1 = V1s2 )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s2 ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V2f1 @ V4x ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ V3f2 @ V4x ) ) ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V2f1 ) @ V0s1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V3f2 ) @ V1s2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__ZERO,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V1s ) )
= fo__c_2Enum_2E0 )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ V0f @ V2x ) )
= fo__c_2Enum_2E0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EABS__DIFF__SUM__IMAGE,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( p
@ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ c_2Earithmetic_2EABS__DIFF @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V1g ) @ V2s ) ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Earithmetic_2EABS__DIFF @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
@ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__MONO__LESS__EQ,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V1g ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__MONO__LESS,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0f @ V3x ) ) @ ( ap @ V1g @ V3x ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V2s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0f @ V4x ) ) @ ( ap @ V1g @ V4x ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V0f ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V1g ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__INJ__o,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V0s ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27b @ A_27a ) @ V1g ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V1g ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ty_2Enum_2Enum @ A_27a ) @ V2f ) @ V1g ) ) @ V0s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__IMAGE__PERMUTES,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27a ) @ V1g ) @ V0s ) @ V0s ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ty_2Enum_2Enum @ A_27a ) @ V2f ) @ V1g ) ) @ V0s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ A_27a ) @ V2f ) @ V0s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ESUM__SET__DEF,axiom,
( c_2Epred__set_2ESUM__SET
= ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ty_2Enum_2Enum ) @ ( c_2Ecombin_2EI @ ty_2Enum_2Enum ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__SET__THM,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
= fo__c_2Enum_2E0 )
& ! [V0x: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2ESUM__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Epred__set_2ESUM__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Enum_2Enum ) @ V1s ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__SET__EMPTY,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2ESUM__SET @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Epred__set_2ESUM__SET__SING,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2ESUM__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) )
= V0n ) ).
thf(conj_thm_2Epred__set_2ESUM__SET__SUBSET__LE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ V0s ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Epred__set_2ESUM__SET @ V0s ) ) @ ( ap @ c_2Epred__set_2ESUM__SET @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__SET__IN__LE,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Epred__set_2ESUM__SET @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__SET__DELETE,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ! [V1e: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2ESUM__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Enum_2Enum ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ V1e ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1e ) ) @ V0s ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Epred__set_2ESUM__SET @ V0s ) ) @ ( inj__ty_2Enum_2Enum @ V1e ) ) ) @ ( ap @ c_2Epred__set_2ESUM__SET @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUM__SET__UNION,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1t ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2ESUM__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ V0s ) @ V1t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Epred__set_2ESUM__SET @ V0s ) ) @ ( ap @ c_2Epred__set_2ESUM__SET @ V1t ) ) ) @ ( ap @ c_2Epred__set_2ESUM__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ ty_2Enum_2Enum ) @ V0s ) @ V1t ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EPROD__IMAGE__DEF,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2EPROD__IMAGE @ A_27a ) @ V0f ) @ V1s ) )
= ( surj__ty_2Enum_2Enum
@ ( ap
@ ( ap
@ ( ap @ ( c_2Epred__set_2EITSET @ A_27a @ ty_2Enum_2Enum )
@ ( lam @ A_27a
@ ^ [V2e: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V3acc: $i] : ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ V2e ) ) @ V3acc ) ) ) )
@ V1s )
@ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPROD__IMAGE__THM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2EPROD__IMAGE @ A_27a ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2EPROD__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ V1e ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPROD__IMAGE @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V2s ) @ V1e ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EPROD__SET__DEF,axiom,
( c_2Epred__set_2EPROD__SET
= ( ap @ ( c_2Epred__set_2EPROD__IMAGE @ ty_2Enum_2Enum ) @ ( c_2Ecombin_2EI @ ty_2Enum_2Enum ) ) ) ).
thf(conj_thm_2Epred__set_2EPROD__SET__THM,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EPROD__SET @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
& ! [V0x: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EPROD__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( ap @ c_2Epred__set_2EPROD__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ty_2Enum_2Enum ) @ V1s ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPROD__SET__EMPTY,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EPROD__SET @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPROD__SET__IMAGE__REDUCTION,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1s ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1s ) ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EPROD__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2x ) @ V1s ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ V0f @ V2x ) ) @ ( ap @ c_2Epred__set_2EPROD__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ty_2Enum_2Enum ) @ V0f ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EMAX__SET__DEF,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2EMAX__SET @ V0s ) ) @ V0s ) )
& ! [V1y: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1y ) ) @ ( ap @ c_2Epred__set_2EMAX__SET @ V0s ) ) ) ) ) )
& ( ( V0s
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMAX__SET @ V0s ) )
= fo__c_2Enum_2E0 ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EMAX__SET__THM,axiom,
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMAX__SET @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
= fo__c_2Enum_2E0 )
& ! [V0e: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMAX__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0e ) ) @ V1s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMAX @ ( inj__ty_2Enum_2Enum @ V0e ) ) @ ( ap @ c_2Epred__set_2EMAX__SET @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EMAX__SET__REWRITES,axiom,
! [V0e: tp__ty_2Enum_2Enum] :
( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMAX__SET @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
= fo__c_2Enum_2E0 )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMAX__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0e ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) )
= V0e ) ) ).
thf(conj_thm_2Epred__set_2EMAX__SET__ELIM,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0P ) )
& ( ( V0P
= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
=> ( p @ ( ap @ V1Q @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
& ! [V2x: tp__ty_2Enum_2Enum] :
( ( ! [V3y: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3y ) ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V3y ) ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0P ) ) )
=> ( p @ ( ap @ V1Q @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) ) )
=> ( p @ ( ap @ V1Q @ ( ap @ c_2Epred__set_2EMAX__SET @ V0P ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EMIN__SET__DEF,axiom,
c_2Epred__set_2EMIN__SET = c_2Ewhile_2ELEAST ).
thf(conj_thm_2Epred__set_2EMIN__SET__ELIM,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( V0P
!= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
& ! [V2x: tp__ty_2Enum_2Enum] :
( ( ! [V3y: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3y ) ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ ( inj__ty_2Enum_2Enum @ V3y ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2x ) ) @ V0P ) ) )
=> ( p @ ( ap @ V1Q @ ( inj__ty_2Enum_2Enum @ V2x ) ) ) ) )
=> ( p @ ( ap @ V1Q @ ( ap @ c_2Epred__set_2EMIN__SET @ V0P ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EMIN__SET__THM,axiom,
( ! [V0e: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMIN__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0e ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) ) )
= V0e )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2e1: tp__ty_2Enum_2Enum,V3e2: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMIN__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2e1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3e2 ) ) @ V1s ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( inj__ty_2Enum_2Enum @ V2e1 ) ) @ ( ap @ c_2Epred__set_2EMIN__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V3e2 ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EMIN__SET__LEM,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2EMIN__SET @ V0s ) ) @ V0s ) )
& ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Epred__set_2EMIN__SET @ V0s ) ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__MIN__SET,axiom,
! [A_27a: del,V0I: $i] :
( ( mem @ V0I @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1J: $i] :
( ( mem @ V1J @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V2n: $i] :
( ( mem @ V2n @ A_27a )
=> ( ( ( V0I
!= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
& ( V1J
!= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ V0I ) @ V1J ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Epred__set_2EMIN__SET @ V1J ) ) @ ( ap @ c_2Epred__set_2EMIN__SET @ V0I ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__MAX__SET,axiom,
! [V0I: $i] :
( ( mem @ V0I @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1J: $i] :
( ( mem @ V1J @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0I ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1J ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ty_2Enum_2Enum ) @ V0I ) @ V1J ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Epred__set_2EMAX__SET @ V0I ) ) @ ( ap @ c_2Epred__set_2EMAX__SET @ V1J ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EMIN__SET__LEQ__MAX__SET,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( V0s
!= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Epred__set_2EMIN__SET @ V0s ) ) @ ( ap @ c_2Epred__set_2EMAX__SET @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EMIN__SET__UNION,axiom,
! [V0A: $i] :
( ( mem @ V0A @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0A ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1B ) )
& ( V0A
!= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
& ( V1B
!= ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMIN__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ V0A ) @ V1B ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( ap @ c_2Epred__set_2EMIN__SET @ V0A ) ) @ ( ap @ c_2Epred__set_2EMIN__SET @ V1B ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EMAX__SET__UNION,axiom,
! [V0A: $i] :
( ( mem @ V0A @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0A ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V1B ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2EMAX__SET @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ V0A ) @ V1B ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMAX @ ( ap @ c_2Epred__set_2EMAX__SET @ V0A ) ) @ ( ap @ c_2Epred__set_2EMAX__SET @ V1B ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EPOW__DEF,axiom,
! [A_27a: del,V0set: $i] :
( ( mem @ V0set @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0set )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V0set ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__POW,axiom,
! [A_27a: del,V0set: $i] :
( ( mem @ V0set @ ( arr @ A_27a @ bool ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1e ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0set ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1e ) @ V0set ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNIV__FUN__TO__BOOL,axiom,
! [A_27a: del] :
( ( c_2Epred__set_2EUNIV @ ( arr @ A_27a @ bool ) )
= ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__POW,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s1 ) @ V1s2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s1 ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V1s2 ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__INSERT__RIGHT,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s1 ) @ V2s2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s1 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) @ V2s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__DELETE__BOTH,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s1 ) @ V1s2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s1 ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1s2 ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPOW__EMPTY,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s )
!= ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPOW__INSERT,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V1s ) ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPOW__EQNS,axiom,
! [A_27a: del] :
( ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) )
& ! [V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) @ V1s ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ bool )
@ ^ [V2ps: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) ) @ V2ps ) ) @ V2ps ) ) )
@ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__POW,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__POW,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EEXP @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__F,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V0x ) @ c_2Ebool_2EF ) ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__T,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V0x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V0x ) @ c_2Ebool_2ET ) ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__ID,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ ( 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 @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0y ) ) ) )
= V0y ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__EQ,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ( ( 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 @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) @ V0y ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0y ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__EQ2,axiom,
! [A_27a: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ( ( 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 @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V0y ) @ V1x ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0y ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__F__COND,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ~ ( p @ ( ap @ ( c_2Epair_2ESND @ A_27b @ bool ) @ ( ap @ V0f @ V1x ) ) ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27a ) @ V0f )
= ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__AND,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( 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 @ V0P @ V2x ) ) @ ( ap @ V1Q @ V2x ) ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ V0P @ V3x ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ V1Q @ V4x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EGSPEC__OR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ( 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 @ V0P @ V2x ) ) @ ( ap @ V1Q @ V2x ) ) ) ) )
= ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3x ) @ ( ap @ V0P @ V3x ) ) ) ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ V1Q @ V4x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Eequiv__on__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Eequiv__on @ A_27a ) @ V0R ) @ V1s ) )
<=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V2x ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V1s ) ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V3x ) ) ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ! [V7z: $i] :
( ( mem @ V7z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6y ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7z ) @ V1s ) )
& ( p @ ( ap @ ( ap @ V0R @ V5x ) @ V6y ) )
& ( p @ ( ap @ ( ap @ V0R @ V6y ) @ V7z ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V5x ) @ V7z ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Epartition__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2t: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V2t )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ bool ) ) @ V2t )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4y ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V1s ) ) @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIGUNION__partition,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Eequiv__on @ A_27a ) @ V0R ) @ V1s ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) )
= V1s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEMPTY__NOT__IN__partition,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Eequiv__on @ A_27a ) @ V0R ) @ V1s ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epartition__elements__disjoint,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Eequiv__on @ A_27a ) @ V0R ) @ V1s ) )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( arr @ A_27a @ bool ) )
=> ! [V3t2: $i] :
( ( mem @ V3t2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t1 ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3t2 ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) ) )
& ( V2t1 != V3t2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V2t1 ) @ V3t2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epartition__elements__interrelate,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Eequiv__on @ A_27a ) @ V0R ) @ V1s ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epartition__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__partition,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V2t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epartition__CARD,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Eequiv__on @ A_27a ) @ V0R ) @ V1s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1s ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epred__set_2ESUM__IMAGE @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2ECARD @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2Epartition @ A_27a ) @ V0R ) @ V1s ) ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Epairwise__def,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Epairwise @ A_27a ) @ V0P ) @ V1s ) )
<=> ! [V2e1: $i] :
( ( mem @ V2e1 @ A_27a )
=> ! [V3e2: $i] :
( ( mem @ V3e2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e1 ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3e2 ) @ V1s ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V2e1 ) @ V3e2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epairwise__UNION,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Epairwise @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s1 ) @ V2s2 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Epairwise @ A_27a ) @ V0R ) @ V1s1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Epairwise @ A_27a ) @ V0R ) @ V2s2 ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s1 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2s2 ) ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V3x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epairwise__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Epairwise @ A_27a ) @ V0R ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2Epairwise @ A_27a ) @ V0R ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EKoenigsLemma,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( 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 @ V0R @ V1x ) @ V2y ) ) ) ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ~ ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4y ) @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V3x ) @ V4y ) ) ) ) ) )
=> ? [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= V3x )
& ! [V6n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ V5f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EKoenigsLemma__WF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( 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 @ V0R @ V1x ) @ V2y ) ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p
@ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4y ) @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESET__EQ__SUBSET,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ( ( V0s1 = V1s2 )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s1 ) @ V1s2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s2 ) @ V0s1 ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__EQN,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s1 ) @ V1s2 ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s1 ) @ V1s2 ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s2 ) @ V0s1 ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__SUBSET__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_2ESUBSET @ A_27a ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__PSUBSET__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_2EPSUBSET @ A_27a ) @ V1t ) @ V2u ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V0s ) @ V2u ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECROSS__EQNS,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V2s2 )
= ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) )
& ( ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0a ) @ V1s1 ) ) @ V2s2 )
= ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ A_27b
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0a ) @ V3y ) ) )
@ V2s2 ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V1s1 ) @ V2s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecount__EQN,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ ty_2Enum_2Enum @ bool ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( c_2Epred__set_2EEMPTY @ ty_2Enum_2Enum ) )
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1p: $i] : ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ V1p ) @ ( ap @ c_2Epred__set_2Ecount @ V1p ) ) ) )
@ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNIQUE__MEMBER__SING,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 ) )
& ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ V1s ) )
=> ( V0x = V2y ) ) ) )
<=> ( V1s
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Einj__surj,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) )
=> ( ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ? [V3f_27: $i] :
( ( mem @ V3f_27 @ ( arr @ A_27b @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27b @ A_27a ) @ V3f_27 ) @ V2t ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Einfinite__rest,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Echooser__def,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2Echooser @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epred__set_2Echooser @ A_27a ) @ V1s ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2Echooser @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V1s ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Echooser__def__compute,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2Echooser @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
= ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epred__set_2Echooser @ A_27a ) @ V1s ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2Echooser @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V1s ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ! [V4n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Epred__set_2Echooser @ A_27a ) @ V3s ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2Echooser @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V3s ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Einfinite__num__inj,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
<=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Ecountable__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ ty_2Enum_2Enum ) @ V1f ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecountable__image__nats,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecountable__surj,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Enum__countable,axiom,
p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ).
thf(conj_thm_2Epred__set_2Esubset__countable,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1t ) @ V0s ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Eimage__countable,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNTABLE__ALT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
=> ? [V3n: tp__ty_2Enum_2Enum] :
( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) )
= V2x ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNTABLE__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 ) @ V0s ) @ V1t ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Efinite__countable,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNTABLE__COUNT,axiom,
! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNTABLE__NUM,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ty_2Enum_2Enum ) @ V0s ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNTABLE__IMAGE__NUM,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V0f ) @ V1s ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Enum__to__pair__def,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2Enum__to__pair @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Enumpair_2Enfst @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Enumpair_2Ensnd @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2Epair__to__num__def,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2Epair__to__num @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Enumpair_2Enpair @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epair__to__num__formula,axiom,
! [V0x: tp__ty_2Enum_2Enum,V1y: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2Epair__to__num @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2EDIV @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V1y ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epair__to__num__inv,axiom,
( ! [V0x: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2Epair__to__num @ ( ap @ c_2Epred__set_2Enum__to__pair @ ( inj__ty_2Enum_2Enum @ V0x ) ) ) )
= V0x )
& ! [V1x: tp__ty_2Enum_2Enum,V2y: tp__ty_2Enum_2Enum] :
( ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ c_2Epred__set_2Enum__to__pair @ ( ap @ c_2Epred__set_2Epair__to__num @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) )
= ( surj__c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( inj__ty_2Enum_2Enum @ V2y ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecross__countable,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Einter__countable,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
| ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Einj__countable,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ V2t ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ebigunion__countable,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ V0s ) )
& ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1x ) @ V0s ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1x ) ) ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Eunion__countable,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Eunion__countable__IFF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Einj__image__countable__IFF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Epow__no__surj,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ~ ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2ESURJ @ A_27a @ ( arr @ A_27a @ bool ) ) @ V1f ) @ V0s ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Einfinite__pow__uncountable,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ~ ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPOW @ A_27a ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecountable__Usum,axiom,
! [A_27a: del,A_27b: del] :
( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecountable__EMPTY,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2Ecountable__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0x ) @ V1s ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecross__countable__IFF,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) )
<=> ( ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ( V1t
= ( c_2Epred__set_2EEMPTY @ A_27b ) )
| ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecountable__Uprod,axiom,
! [A_27a: del,A_27b: del] :
( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EEXPLICIT__ENUMERATE__MONO,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EREST @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1s ) ) @ V1s ) ) ) ).
thf(conj_thm_2Epred__set_2EEXPLICIT__ENUMERATE__NOT__EMPTY,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EREST @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ V1s )
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINFINITE__EXPLICIT__ENUMERATE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
=> ( p
@ ( ap
@ ( ap
@ ( ap @ ( c_2Epred__set_2EINJ @ ty_2Enum_2Enum @ A_27a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1n: $i] : ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Earithmetic_2EFUNPOW @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EREST @ A_27a ) ) @ V1n ) @ V0s ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) )
@ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EBIJ__NUM__COUNTABLE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) ) ) ) ).
thf(ax_thm_2Epred__set_2Eenumerate__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2Eenumerate @ A_27a ) @ V0s )
= ( ap @ ( c_2Emin_2E_40 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
@ ^ [V1f: $i] : ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EENUMERATE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ A_27a ) @ ( ap @ ( c_2Epred__set_2Eenumerate @ A_27a ) @ V0s ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNTABLE__ALT__BIJ,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0s ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
| ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ ty_2Enum_2Enum @ A_27a ) @ ( ap @ ( c_2Epred__set_2Eenumerate @ A_27a ) @ V0s ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) @ V0s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECOUNTABLE__ENUM,axiom,
! [A_27a: del,V0c: $i] :
( ( mem @ V0c @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2Ecountable @ A_27a ) @ V0c ) )
<=> ( ( V0c
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
| ? [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ( V0c
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ A_27a ) @ V1f ) @ ( c_2Epred__set_2EUNIV @ ty_2Enum_2Enum ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__BIGUNION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1M ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) ) @ V1M ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__DIFF,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ! [V2s3: $i] :
( ( mem @ V2s3 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s1 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s2 ) @ V2s3 ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s1 ) @ V1s2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ V0s1 ) @ V2s3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__SUBSET__EQN,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ( ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0A ) @ V1B )
= V0A )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0A ) @ V1B ) ) )
& ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0A ) @ V1B )
= V1B )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1B ) @ V0A ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPSUBSET__SING,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ A_27a )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
<=> ( V1x
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINTER__UNION,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0A ) @ V1B ) ) @ V0A )
= V0A )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1B ) @ V0A ) ) @ V0A )
= V0A )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0A ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0A ) @ V1B ) )
= V0A )
& ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0A ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1B ) @ V0A ) )
= V0A ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__DELETE,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0A ) @ V1B ) ) @ V2x )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0A ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V1B ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDELETE__SUBSET__INSERT,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1e: $i] :
( ( mem @ V1e @ A_27a )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V1e ) ) @ V2s2 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1e ) @ V2s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__INSERT__EXPAND,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1y ) @ V2P ) ) )
<=> ( ( V0x = V1y )
| ( ( V0x != V1y )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2P ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__INTER,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( arr @ A_27a @ bool ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s1 ) )
| ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s2 ) ) )
=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s1 ) @ V1s2 ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EINSERT__EQ__SING,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2y ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( ( V1x = V2y )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2y ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__UNION__LE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1t ) ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__SUBSET__gen,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2u: $i] :
( ( mem @ V2u @ ( arr @ A_27a @ bool ) )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2u ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2u ) ) @ V3t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ECARD__REST,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( V0s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2ESUBSET__DIFF__EMPTY,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 )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EDIFF__INTER__SUBSET,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ bool ) )
=> ! [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 ) @ V0r ) @ V1s ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0r ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EUNION__DIFF__2,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 ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V0s ) @ V1t ) )
= V0s ) ) ) ).
thf(conj_thm_2Epred__set_2Ecount__add,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ty_2Enum_2Enum ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__EQ__SING,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ A_27a ) @ V0f ) @ V1s )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V2z ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
<=> ( ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27b ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ V1s ) )
=> ( ( ap @ V0f @ V3x )
= V2z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecount__add1,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( ap @ c_2Epred__set_2Ecount @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ecompl__insert,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V0s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V0s ) ) @ V1x ) ) ) ) ).
thf(conj_thm_2Epred__set_2Ein__max__set,axiom,
! [V0s: $i] :
( ( mem @ V0s @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Enum_2Enum ) @ V0s ) )
=> ! [V1x: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V1x ) ) @ ( ap @ c_2Epred__set_2EMAX__SET @ V0s ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EPREIMAGE__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s )
= ( 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_2EIN @ A_27b ) @ ( ap @ V0f @ V2x ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__ALT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ A_27b ) @ V1s ) @ V0f ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__PREIMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V2x ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__applied,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) @ V2x ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V2x ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__EMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__UNIV,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ A_27b ) )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__COMPL,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27b ) @ V1s ) )
= ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__UNION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V1s ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__INTER,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V1s ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__BIGUNION,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27b ) @ V1s ) )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( arr @ A_27b @ bool ) @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__COMP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27c ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27c @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27c ) @ V1g ) @ V2s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V1g ) @ V0f ) ) @ V2s ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__DIFF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27b ) @ V1s ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__I,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ bool ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__K,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27b @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27b ) @ V0x ) ) @ V1s )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( arr @ A_27b @ bool ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) @ ( c_2Epred__set_2EEMPTY @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__DISJOINT,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27b ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__CROSS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ ( arr @ A_27b @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ ( arr @ A_27c @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2ECROSS @ A_27b @ A_27c ) @ V1a ) @ V2b ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27b @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epair_2EFST @ A_27b @ A_27c ) ) @ V0f ) ) @ V1a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ ( c_2Epair_2ESND @ A_27b @ A_27c ) ) @ V0f ) ) @ V2b ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__COMPL__INTER,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ! [V2sp: $i] :
( ( mem @ V2sp @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27b ) @ V1t ) ) ) @ V2sp )
= ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V2sp ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1t ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EPREIMAGE__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIMAGE__PREIMAGE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EPREIMAGE @ A_27a @ A_27b ) @ V0f ) @ V1s ) ) ) @ V1s ) ) ) ) ).
thf(ax_thm_2Epred__set_2Eis__measure__maximal__def,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2Eis__measure__maximal @ A_27a ) @ V0m ) @ V1s ) @ V2x ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ V0m @ V3y ) ) @ ( ap @ V0m @ V2x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EFINITE__is__measure__maximal,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2Eis__measure__maximal @ A_27a ) @ V0m ) @ V1s ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Eis__measure__maximal__SING,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2Eis__measure__maximal @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V2y ) )
<=> ( V2y = V1x ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2Eis__measure__maximal__INSERT,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2m: $i] :
( ( mem @ V2m @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V2m @ V3e ) ) @ ( ap @ V2m @ V0x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2Eis__measure__maximal @ A_27a ) @ V2m ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3e ) @ V1s ) ) @ V4y ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2Eis__measure__maximal @ A_27a ) @ V2m ) @ V1s ) @ V4y ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------