TPTP Problem File: ITP004+5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP004+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Epred__set_2EREST__SUBSET.p, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Epred__set_2EREST__SUBSET.p [Gau20]
% : HL401501+5.p [TPAP]
% Status : Theorem
% Rating : 0.83 v8.2.0, 0.78 v7.5.0
% Syntax : Number of formulae : 2447 ( 193 unt; 0 def)
% Number of atoms : 12453 (1816 equ)
% Maximal formula atoms : 94 ( 5 avg)
% Number of connectives : 10298 ( 292 ~; 236 |;1326 &)
% ( 814 <=>;7630 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 8 avg)
% Maximal term depth : 16 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 621 ( 621 usr; 136 con; 0-7 aty)
% Number of variables : 7474 (7232 !; 242 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
include('Axioms/ITP001/ITP002+5.ax').
include('Axioms/ITP001/ITP003+5.ax').
include('Axioms/ITP001/ITP004+5.ax').
include('Axioms/ITP001/ITP007+5.ax').
include('Axioms/ITP001/ITP006+5.ax').
include('Axioms/ITP001/ITP005+5.ax').
include('Axioms/ITP001/ITP008+5.ax').
include('Axioms/ITP001/ITP009+5.ax').
include('Axioms/ITP001/ITP010+5.ax').
include('Axioms/ITP001/ITP012+5.ax').
include('Axioms/ITP001/ITP011+5.ax').
include('Axioms/ITP001/ITP013+5.ax').
include('Axioms/ITP001/ITP014+5.ax').
include('Axioms/ITP001/ITP015+5.ax').
include('Axioms/ITP001/ITP017+5.ax').
include('Axioms/ITP001/ITP016+5.ax').
include('Axioms/ITP001/ITP019+5.ax').
include('Axioms/ITP001/ITP018+5.ax').
include('Axioms/ITP001/ITP021+5.ax').
include('Axioms/ITP001/ITP022+5.ax').
include('Axioms/ITP001/ITP020+5.ax').
include('Axioms/ITP001/ITP024+5.ax').
include('Axioms/ITP001/ITP023+5.ax').
include('Axioms/ITP001/ITP025+5.ax').
include('Axioms/ITP001/ITP026+5.ax').
%------------------------------------------------------------------------------
fof(mem_c_2Epred__set_2EBIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EBIGINTER(A_27a),arr(arr(arr(A_27a,bool),bool),arr(A_27a,bool))) ) ).
fof(mem_c_2Epred__set_2EBIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EBIGUNION(A_27a),arr(arr(arr(A_27a,bool),bool),arr(A_27a,bool))) ) ).
fof(mem_c_2Epred__set_2EBIJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2ECARD,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2ECARD(A_27a),arr(arr(A_27a,bool),ty_2Enum_2Enum)) ) ).
fof(mem_c_2Epred__set_2ECHOICE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2ECHOICE(A_27a),arr(arr(A_27a,bool),A_27a)) ) ).
fof(mem_c_2Epred__set_2ECOMPL,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2ECOMPL(A_27a),arr(arr(A_27a,bool),arr(A_27a,bool))) ) ).
fof(mem_c_2Epred__set_2ECROSS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2EDELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EDELETE(A_27a),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool)))) ) ).
fof(mem_c_2Epred__set_2EDFUNSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2EDIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EDIFF(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) ).
fof(mem_c_2Epred__set_2EDISJOINT,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EDISJOINT(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Epred__set_2EEMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EEMPTY(A_27a),arr(A_27a,bool)) ) ).
fof(mem_c_2Epred__set_2EFINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EFINITE(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(mem_c_2Epred__set_2EFUNSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2EGSPEC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epred__set_2EGSPEC(A_27a,A_27b),arr(arr(A_27b,ty_2Epair_2Eprod(A_27a,bool)),arr(A_27a,bool))) ) ) ).
fof(mem_c_2Epred__set_2EIMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epred__set_2EIMAGE(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(A_27b,bool)))) ) ) ).
fof(mem_c_2Epred__set_2EINJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2EINSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EINSERT(A_27a),arr(A_27a,arr(arr(A_27a,bool),arr(A_27a,bool)))) ) ).
fof(mem_c_2Epred__set_2EINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EINTER(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) ).
fof(mem_c_2Epred__set_2EITSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2ELINV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2ELINV__OPT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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))))) ) ) ).
fof(mem_c_2Epred__set_2EMAX__SET,axiom,
mem(c_2Epred__set_2EMAX__SET,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) ).
fof(mem_c_2Epred__set_2EMIN__SET,axiom,
mem(c_2Epred__set_2EMIN__SET,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) ).
fof(mem_c_2Epred__set_2EPOW,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EPOW(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Epred__set_2EPREIMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epred__set_2EPREIMAGE(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,bool),arr(A_27a,bool)))) ) ) ).
fof(mem_c_2Epred__set_2EPROD__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EPROD__IMAGE(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(arr(A_27a,bool),ty_2Enum_2Enum))) ) ).
fof(mem_c_2Epred__set_2EPROD__SET,axiom,
mem(c_2Epred__set_2EPROD__SET,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) ).
fof(mem_c_2Epred__set_2EPSUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EPSUBSET(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Epred__set_2EREL__RESTRICT,axiom,
! [A_27a] :
( ne(A_27a)
=> 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))))) ) ).
fof(mem_c_2Epred__set_2EREST,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EREST(A_27a),arr(arr(A_27a,bool),arr(A_27a,bool))) ) ).
fof(mem_c_2Epred__set_2ERINV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2ESING,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2ESING(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(mem_c_2Epred__set_2ESUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2ESUBSET(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Epred__set_2ESUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2ESUM__IMAGE(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(arr(A_27a,bool),ty_2Enum_2Enum))) ) ).
fof(mem_c_2Epred__set_2ESUM__SET,axiom,
mem(c_2Epred__set_2ESUM__SET,arr(arr(ty_2Enum_2Enum,bool),ty_2Enum_2Enum)) ).
fof(mem_c_2Epred__set_2ESURJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)))) ) ) ).
fof(mem_c_2Epred__set_2EUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EUNION(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) ).
fof(mem_c_2Epred__set_2EUNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EUNIV(A_27a),arr(A_27a,bool)) ) ).
fof(mem_c_2Epred__set_2Echooser,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2Echooser(A_27a),arr(arr(A_27a,bool),arr(ty_2Enum_2Enum,A_27a))) ) ).
fof(mem_c_2Epred__set_2Ecount,axiom,
mem(c_2Epred__set_2Ecount,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool))) ).
fof(mem_c_2Epred__set_2Ecountable,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2Ecountable(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(mem_c_2Epred__set_2Eenumerate,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2Eenumerate(A_27a),arr(arr(A_27a,bool),arr(ty_2Enum_2Enum,A_27a))) ) ).
fof(mem_c_2Epred__set_2Eequiv__on,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2Eequiv__on(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Epred__set_2Eis__measure__maximal,axiom,
! [A_27a] :
( ne(A_27a)
=> 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)))) ) ).
fof(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))) ).
fof(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)) ).
fof(mem_c_2Epred__set_2Epairwise,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2Epairwise(A_27a),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Epred__set_2Epartition,axiom,
! [A_27a] :
( ne(A_27a)
=> 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)))) ) ).
fof(mem_c_2Epred__set_2Eschroeder__close,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2Eschroeder__close(A_27a),arr(arr(A_27a,A_27a),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) ).
fof(conj_thm_2Epred__set_2ESPECIFICATION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0P))
<=> p(ap(V0P,V1x)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__APP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1P))
<=> p(ap(V1P,V0x)) ) ) ) ) ).
fof(lameq_f252,axiom,
! [A_27a,V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2x] : ap(f252(A_27a,V1P),V2x) = ap(V1P,V2x) ) ).
fof(conj_thm_2Epred__set_2EIN__ABS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),f252(A_27a,V1P)))
<=> p(ap(V1P,V0x)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EEXTENSION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( V0s = V1t
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
<=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ENOT__EQUAL__SETS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( V0s != V1t
<=> ? [V2x] :
( mem(V2x,A_27a)
& ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t))
<=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ENUM__SET__WOP,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ( ? [V1n] :
( mem(V1n,ty_2Enum_2Enum)
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1n),V0s)) )
<=> ? [V2n] :
( mem(V2n,ty_2Enum_2Enum)
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2n),V0s))
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3m),V0s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V2n),V3m)) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EGSPECIFICATION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27b,ty_2Epair_2Eprod(A_27a,bool)))
=> ! [V1v] :
( mem(V1v,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1v),ap(c_2Epred__set_2EGSPEC(A_27a,A_27b),V0f)))
<=> ? [V2x] :
( mem(V2x,A_27b)
& ap(ap(c_2Epair_2E_2C(A_27a,bool),V1v),c_2Ebool_2ET) = ap(V0f,V2x) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EGSPECIFICATION__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27b,ty_2Epair_2Eprod(A_27a,bool)))
=> ! [V1v] :
( mem(V1v,A_27a)
=> ( p(ap(ap(c_2Epred__set_2EGSPEC(A_27a,A_27b),V0f),V1v))
<=> ? [V2x] :
( mem(V2x,A_27b)
& ap(ap(c_2Epair_2E_2C(A_27a,bool),V1v),c_2Ebool_2ET) = ap(V0f,V2x) ) ) ) ) ) ) ).
fof(lameq_f253,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] : ap(f253(A_27a,V0P),V1x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1x),ap(V0P,V1x)) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__ETA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f253(A_27a,V0P)) = V0P ) ) ).
fof(lameq_f254,axiom,
! [A_27a,A_27b,V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] : ap(f254(A_27a,A_27b,V0P,V1x),V2y) = 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)) ) ) ).
fof(lameq_f255,axiom,
! [A_27b,A_27a,V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x] : ap(f255(A_27b,A_27a,V0P),V1x) = f254(A_27a,A_27b,V0P,V1x) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__PAIR__ETA,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( 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)),f255(A_27b,A_27a,V0P))) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),V0P) ) ) ) ).
fof(lameq_f256,axiom,
! [A_27a,V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2x] : ap(f256(A_27a,V1P),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(V1P,V2x)) ) ).
fof(conj_thm_2Epred__set_2EIN__GSPEC__IFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0y] :
( mem(V0y,A_27a)
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0y),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f256(A_27a,V1P))))
<=> p(ap(V1P,V0y)) ) ) ) ) ).
fof(lameq_f257,axiom,
! [A_27a,A_27b,V2P] :
( mem(V2P,arr(A_27a,arr(A_27b,bool)))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] : ap(f257(A_27a,A_27b,V2P,V3x),V4y) = 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)) ) ) ).
fof(lameq_f258,axiom,
! [A_27b,A_27a,V2P] :
( mem(V2P,arr(A_27a,arr(A_27b,bool)))
=> ! [V3x] : ap(f258(A_27b,A_27a,V2P),V3x) = f257(A_27a,A_27b,V2P,V3x) ) ).
fof(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__IFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ! [V2P] :
( 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)),f258(A_27b,A_27a,V2P)))))
<=> p(ap(ap(V2P,V0x),V1y)) ) ) ) ) ) ) ).
fof(lameq_f259,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V3P] :
( mem(V3P,arr(A_27a,bool))
=> ! [V4x] : ap(f259(A_27b,A_27a,V0f,V3P),V4x) = ap(ap(c_2Epair_2E_2C(A_27b,bool),ap(V0f,V4x)),ap(V3P,V4x)) ) ) ).
fof(conj_thm_2Epred__set_2EIN__GSPEC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2x] :
( mem(V2x,A_27b)
=> ! [V3P] :
( 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),f259(A_27b,A_27a,V0f,V3P)))) ) ) ) ) ) ) ) ).
fof(lameq_f260,axiom,
! [A_27b,A_27a,V2x] :
( mem(V2x,A_27b)
=> ! [V3P] :
( mem(V3P,arr(A_27a,bool))
=> ! [V4y] : ap(f260(A_27b,A_27a,V2x,V3P),V4y) = 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)) ) ) ).
fof(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1b] :
( mem(V1b,A_27b)
=> ! [V2x] :
( mem(V2x,A_27b)
=> ! [V3P] :
( 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),f260(A_27b,A_27a,V2x,V3P))))
<=> ( p(ap(V3P,V0a))
& V1b = V2x ) ) ) ) ) ) ) ) ).
fof(lameq_f261,axiom,
! [A_27a,A_27b,V2x] :
( mem(V2x,A_27a)
=> ! [V3P] :
( mem(V3P,arr(A_27b,bool))
=> ! [V4y] : ap(f261(A_27a,A_27b,V2x,V3P),V4y) = 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)) ) ) ).
fof(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1b] :
( mem(V1b,A_27b)
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3P] :
( 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),f261(A_27a,A_27b,V2x,V3P))))
<=> ( p(ap(V3P,V1b))
& V0a = V2x ) ) ) ) ) ) ) ) ).
fof(lameq_f262,axiom,
! [A_27a,V2P] :
( mem(V2P,arr(A_27a,bool))
=> ! [V3x] : ap(f262(A_27a,V2P),V3x) = 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)) ) ).
fof(conj_thm_2Epred__set_2EPAIR__IN__GSPEC__same,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1b] :
( mem(V1b,A_27a)
=> ! [V2P] :
( 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),f262(A_27a,V2P))))
<=> ( p(ap(V2P,V0a))
& V0a = V1b ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESET__MINIMUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1M] :
( mem(V1M,arr(A_27a,ty_2Enum_2Enum))
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
& ! [V4y] :
( 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))) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EEMPTY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epred__set_2EEMPTY(A_27a) = k(A_27a,c_2Ebool_2EF) ) ).
fof(conj_thm_2Epred__set_2ENOT__IN__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))) ) ) ).
fof(conj_thm_2Epred__set_2EMEMBER__NOT__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ? [V1x] :
( mem(V1x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) )
<=> V0s != c_2Epred__set_2EEMPTY(A_27a) ) ) ) ).
fof(conj_thm_2Epred__set_2EEMPTY__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ( p(ap(c_2Epred__set_2EEMPTY(A_27a),V0x))
<=> $false ) ) ) ).
fof(ax_thm_2Epred__set_2EUNIV__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epred__set_2EUNIV(A_27a) = k(A_27a,c_2Ebool_2ET) ) ).
fof(conj_thm_2Epred__set_2EIN__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),c_2Epred__set_2EUNIV(A_27a))) ) ) ).
fof(conj_thm_2Epred__set_2EUNIV__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> p(ap(c_2Epred__set_2EUNIV(A_27a),V0x)) ) ) ).
fof(conj_thm_2Epred__set_2EUNIV__NOT__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epred__set_2EUNIV(A_27a) != c_2Epred__set_2EEMPTY(A_27a) ) ).
fof(conj_thm_2Epred__set_2EEMPTY__NOT__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epred__set_2EEMPTY(A_27a) != c_2Epred__set_2EUNIV(A_27a) ) ).
fof(conj_thm_2Epred__set_2EEQ__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) )
<=> V0s = c_2Epred__set_2EUNIV(A_27a) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__EQ__UNIV__IMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( V0s = c_2Epred__set_2EUNIV(A_27a)
=> ! [V1v] :
( mem(V1v,A_27a)
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V1v),V0s)) ) ) ) ) ).
fof(ax_thm_2Epred__set_2ESUBSET__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0P),V1Q))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1Q)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0s,V2x))
=> p(ap(V1t,V2x)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V0s)) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__ANTISYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EEMPTY__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0s)) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),c_2Epred__set_2EUNIV(A_27a))) ) ) ).
fof(conj_thm_2Epred__set_2EUNIV__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EEQ__SUBSET__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__SUBSET__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__ADD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ! [V2d] :
( mem(V2d,ty_2Enum_2Enum)
=> ( ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,V3n)),ap(V0f,ap(c_2Enum_2ESUC,V3n)))) )
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,V1n)),ap(V0f,ap(ap(c_2Earithmetic_2E_2B,V1n),V2d)))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EK__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,bool)
=> ! [V1y] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__K,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y] :
( 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) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EPSUBSET__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__IRREFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ~ p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V0s)) ) ) ).
fof(conj_thm_2Epred__set_2ENOT__PSUBSET__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ~ p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),c_2Epred__set_2EEMPTY(A_27a))) ) ) ).
fof(conj_thm_2Epred__set_2ENOT__UNIV__PSUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ~ p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0s)) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),c_2Epred__set_2EUNIV(A_27a)))
<=> ? [V1x] :
( mem(V1x,A_27a)
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) ) ) ) ) ).
fof(lameq_f263,axiom,
! [A_27a,V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2x] : ap(f263(A_27a,V1t,V0s),V2x) = 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))) ) ) ).
fof(ax_thm_2Epred__set_2EUNION__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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),f263(A_27a,V1t,V0s)) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__ASSOC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__IDEMPOT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V0s) = V0s ) ) ).
fof(conj_thm_2Epred__set_2EUNION__COMM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__UNION__ABSORPTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EUNION(A_27a),c_2Epred__set_2EEMPTY(A_27a)),V0s) = V0s )
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),c_2Epred__set_2EEMPTY(A_27a)) = V1s ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( 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] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EEMPTY__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFORALL__IN__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ! [V3x] :
( 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] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1s))
=> p(ap(V0P,V4x)) ) )
& ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V5x),V2t))
=> p(ap(V0P,V5x)) ) ) ) ) ) ) ) ) ).
fof(lameq_f264,axiom,
! [A_27a,V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2x] : ap(f264(A_27a,V1t,V0s),V2x) = 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))) ) ) ).
fof(ax_thm_2Epred__set_2EINTER__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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),f264(A_27a,V1t,V0s)) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__ASSOC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__IDEMPOT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V0s) = V0s ) ) ).
fof(conj_thm_2Epred__set_2EINTER__COMM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__INTER__ABSORPTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__INTER1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__INTER2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( 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] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EINTER(A_27a),c_2Epred__set_2EUNIV(A_27a)),V0s) = V0s )
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EINTER(A_27a),V1s),c_2Epred__set_2EUNIV(A_27a)) = V1s ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__OVER__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__OVER__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EDISJOINT__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__DISJOINT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1t))
<=> ~ ? [V2x] :
( mem(V2x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__ALT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1t))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__EMPTY__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( V0s = c_2Epred__set_2EEMPTY(A_27a)
<=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V0s)) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__EMPTY__REFL__RWT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V0s))
<=> V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__UNION__BOTH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2u] :
( 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)) ) ) ) ) ) ).
fof(lameq_f265,axiom,
! [A_27a,V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2x] : ap(f265(A_27a,V1t,V0s),V2x) = 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)))) ) ) ).
fof(ax_thm_2Epred__set_2EDIFF__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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),f265(A_27a,V1t,V0s)) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),c_2Epred__set_2EEMPTY(A_27a)) = V0s ) ) ).
fof(conj_thm_2Epred__set_2EEMPTY__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__EQ__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V0s) = c_2Epred__set_2EEMPTY(A_27a) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__DIFF__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y] :
( mem(V1y,arr(A_27a,bool))
=> ! [V2z] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__COMM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y] :
( mem(V1y,arr(A_27a,bool))
=> ! [V2z] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__SAME__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2g] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__INTER2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__DIFFS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0g] :
( mem(V0g,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( ( ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V1f,V4n)),ap(V1f,ap(c_2Enum_2ESUC,V4n)))) )
& ! [V5n] :
( mem(V5n,ty_2Enum_2Enum)
=> ap(V0g,V5n) = ap(ap(c_2Epred__set_2EDIFF(A_27a),ap(V1f,ap(c_2Enum_2ESUC,V5n))),ap(V1f,V5n)) )
& V2m != V3n )
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0g,V2m)),ap(V0g,V3n))) ) ) ) ) ) ) ).
fof(lameq_f266,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V2y] : ap(f266(A_27a,V1s,V0x),V2y) = 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))) ) ) ).
fof(ax_thm_2Epred__set_2EINSERT__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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),f266(A_27a,V1s,V0x)) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2s] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2s] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOMPONENT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2ESET__CASES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( V0s = c_2Epred__set_2EEMPTY(A_27a)
| ? [V1x] :
( mem(V1x,A_27a)
& ? [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDECOMPOSITION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s))
<=> ? [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EABSORPTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EABSORPTION__RWT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__COMM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2ENOT__INSERT__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s) != c_2Epred__set_2EEMPTY(A_27a) ) ) ) ).
fof(conj_thm_2Epred__set_2ENOT__EMPTY__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> c_2Epred__set_2EEMPTY(A_27a) != ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),V1s) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__UNION__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__INSERT_27,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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))) ) ) ) ) ).
fof(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))) ).
fof(conj_thm_2Epred__set_2EFORALL__IN__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ! [V3x] :
( 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] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s))
=> p(ap(V0P,V4x)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EEXISTS__IN__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ? [V3x] :
( 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] :
( mem(V4x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V2s))
& p(ap(V0P,V4x)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EDELETE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( 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 ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( 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 ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__NON__ELEMENT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__NON__ELEMENT__RWT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__DELETE__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2x_27] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EEMPTY__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2EELT__IN__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__COMM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2s] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__INSERT__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__OF__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__INSERT__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V1t))
<=> ? [V2x] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__MEMBER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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] :
( mem(V2y,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),V1t))
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),V0s)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__DELETE__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2ECHOICE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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)) ) ) ) ).
fof(ax_thm_2Epred__set_2EREST__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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)) ) ) ).
fof(conj_thm_2Epred__set_2EIN__REST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EREST__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECHOICE__NOT__IN__REST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ).
fof(conj_thm_2Epred__set_2ECHOICE__INSERT__REST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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 ) ) ) ).
fof(conj_thm_2Epred__set_2EREST__SUBSET,conjecture,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s)),V0s)) ) ) ).
%------------------------------------------------------------------------------