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 : 693 ( 19 unt; 0 def)
% Number of atoms : 3826 ( 489 equ)
% Maximal formula atoms : 22 ( 5 avg)
% Number of connectives : 3239 ( 106 ~; 28 |; 346 &)
% ( 197 <=>;2562 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 175 ( 175 usr; 37 con; 0-5 aty)
% Number of variables : 2432 (2353 !; 79 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
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,axiom,
! [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)) ) ) ).
fof(conj_thm_2Epred__set_2EREST__PSUBSET,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_2EPSUBSET(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s)),V0s)) ) ) ) ).
fof(ax_thm_2Epred__set_2ESING__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2ESING(A_27a),V0s))
<=> ? [V1x] :
( mem(V1x,A_27a)
& V0s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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)))) ) ) ).
fof(conj_thm_2Epred__set_2ESING__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ( p(ap(c_2Epred__set_2ESING(A_27a),c_2Epred__set_2EEMPTY(A_27a)))
<=> $false ) ) ).
fof(conj_thm_2Epred__set_2ESING__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESING__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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 ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESING__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( p(ap(ap(ap(c_2Epred__set_2EINSERT(A_27a),V1y),c_2Epred__set_2EEMPTY(A_27a)),V0x))
<=> V0x = V1y ) ) ) ) ).
fof(conj_thm_2Epred__set_2ENOT__SING__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2ENOT__EMPTY__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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)) ) ) ).
fof(conj_thm_2Epred__set_2EEQUAL__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__SING__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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))) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__SING__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2ESING__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__EQ__SING,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) = c_2Epred__set_2EEMPTY(A_27a)
<=> V0s = ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),c_2Epred__set_2EEMPTY(A_27a)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECHOICE__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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 ) ) ).
fof(conj_thm_2Epred__set_2EREST__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2ESING__IFF__EMPTY__REST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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) ) ) ) ) ).
fof(lameq_f267,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] : ap(f267(A_27b,A_27a,V0f,V1s),V2x) = ap(ap(c_2Epair_2E_2C(A_27b,bool),ap(V0f,V2x)),ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s)) ) ) ).
fof(ax_thm_2Epred__set_2EIMAGE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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),f267(A_27b,A_27a,V0f,V1s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0y] :
( mem(V0y,A_27b)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( 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] :
( mem(V3x,A_27a)
& V0y = ap(V2f,V3x)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0y] :
( mem(V0y,A_27b)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V2f),V1s),V0y))
<=> ? [V3x] :
( mem(V3x,A_27a)
& V0y = ap(V2f,V3x)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__IN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s))
=> ! [V2f] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),i(A_27a)),V0s) = V0s ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__I,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),c_2Ecombin_2EI(A_27a)),V0s) = V0s ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__II,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Epred__set_2EIMAGE(A_27a,A_27a),c_2Ecombin_2EI(A_27a)) = c_2Ecombin_2EI(arr(A_27a,bool)) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__COMPOSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27b,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2s] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2s] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__EQ__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2s] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [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))
=> ! [V2f] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2] :
( mem(V2s2,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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 ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2] :
( mem(V2s2,arr(A_27a,bool))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f_27] :
( mem(V2f_27,arr(A_27a,A_27b))
=> ! [V3s_27] :
( mem(V3s_27,arr(A_27a,bool))
=> ( ( V1s = V3s_27
& ! [V4x] :
( 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) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27c,A_27a))
=> ! [V2s] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFORALL__IN__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ( ! [V3y] :
( 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] :
( mem(V4x,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),V2s))
=> p(ap(V0P,ap(V1f,V4x))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EEXISTS__IN__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27b,bool))
=> ( ? [V3y] :
( 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] :
( mem(V4x,A_27b)
& p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),V2s))
& p(ap(V0P,ap(V1f,V4x))) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EINJ__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t))
<=> ( ! [V3x] :
( 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] :
( mem(V4x,A_27a)
=> ! [V5y] :
( 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 ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__IFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t))
<=> ( ! [V3x] :
( 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] :
( mem(V4x,A_27a)
=> ! [V5y] :
( 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 ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27a),i(A_27a)),V0s),V0s)) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__COMPOSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ! [V4u] :
( 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)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1s] :
( 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] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V0s),V1t))
=> ! [V3e] :
( 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)))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( 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] :
( mem(V4y,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4y),V2s))
& ap(V0f,V1x) = ap(V0f,V4y) )
=> V1x = V4y ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__EXTEND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0b] :
( mem(V0b,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ! [V3s0] :
( mem(V3s0,arr(A_27a,bool))
=> ! [V4t0] :
( 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)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__IMAGE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2ESURJ__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t))
<=> ( ! [V3x] :
( 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] :
( mem(V4x,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4x),V2t))
=> ? [V5y] :
( mem(V5y,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V5y),V1s))
& ap(V0f,V5y) = V4x ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESURJ__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27a),i(A_27a)),V0s),V0s)) ) ) ).
fof(conj_thm_2Epred__set_2ESURJ__COMPOSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ! [V4u] :
( 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)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESURJ__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1s] :
( 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] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__SURJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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 ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESURJ__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESURJ__IMP__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V2f),V0s),V1t)) )
=> ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27a),V3g),V1t),V0s)) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EBIJ__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27a),i(A_27a)),V0s),V0s)) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__IMP__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( 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] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ap(V0f,V1x) = ap(V0f,V2y)
<=> V1x = V2y ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1s] :
( 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] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__COMPOSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( mem(V3t,arr(A_27b,bool))
=> ! [V4u] :
( 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)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s),V1t))
=> ! [V3e] :
( 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)))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__IMAGE__BIJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ? [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__BIJ__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s0] :
( mem(V0s0,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3t] :
( 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))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__SYM__IMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s),V1t)) )
=> ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27a),V3g),V1t),V0s)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s),V1t)) )
<=> ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27a),V3g),V1t),V0s)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27c,bool))
=> ! [V2u] :
( mem(V2u,arr(A_27b,bool))
=> ( ( ? [V3f] :
( mem(V3f,arr(A_27a,A_27c))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27c),V3f),V0s),V1t)) )
& ? [V4g] :
( mem(V4g,arr(A_27c,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27c,A_27b),V4g),V1t),V2u)) ) )
=> ? [V5h] :
( mem(V5h,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V5h),V0s),V2u)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__INV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t))
=> ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27a),V3g),V2t),V1s))
& ! [V4x] :
( 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] :
( 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 ) ) ) ) ) ) ) ) ) ).
fof(lameq_f268,axiom,
! [A_27b,A_27a,V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V3x] : ap(f268(A_27b,A_27a,V1Q,V2f,V0P),V3x) = 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)) ) ) ) ).
fof(lameq_f269,axiom,
! [A_27b,A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ! [V2f] : ap(f269(A_27b,A_27a,V0P,V1Q),V2f) = ap(c_2Ebool_2E_21(A_27a),f268(A_27b,A_27a,V1Q,V2f,V0P)) ) ) ).
fof(ax_thm_2Epred__set_2EFUNSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V0P),V1Q) = f269(A_27b,A_27a,V0P,V1Q) ) ) ) ) ).
fof(lameq_f270,axiom,
! [A_27b,A_27a,V1Q] :
( mem(V1Q,arr(A_27a,arr(A_27b,bool)))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V3x] : ap(f270(A_27b,A_27a,V1Q,V2f,V0P),V3x) = 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))) ) ) ) ).
fof(lameq_f271,axiom,
! [A_27b,A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,arr(A_27b,bool)))
=> ! [V2f] : ap(f271(A_27b,A_27a,V0P,V1Q),V2f) = ap(c_2Ebool_2E_21(A_27a),f270(A_27b,A_27a,V1Q,V2f,V0P)) ) ) ).
fof(ax_thm_2Epred__set_2EDFUNSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,arr(A_27b,bool)))
=> ap(ap(c_2Epred__set_2EDFUNSET(A_27a,A_27b),V0P),V1Q) = f271(A_27b,A_27a,V0P,V1Q) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__FUNSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2Q] :
( 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] :
( 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)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFUNSET__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2Q] :
( mem(V2Q,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2EFUNSET(A_27a,A_27b),V1P),V2Q),V0f))
<=> ! [V3x] :
( 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)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__DFUNSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2Q] :
( 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] :
( 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))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDFUNSET__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2Q] :
( 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] :
( 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))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFUNSET__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3x] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNIV__FUNSET__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)) ) ) ).
fof(conj_thm_2Epred__set_2EFUNSET__DFUNSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1y] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EEMPTY__FUNSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EFUNSET__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFUNSET__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,arr(A_27a,bool))
=> ! [V1b] :
( mem(V1b,arr(A_27b,bool))
=> ! [V2c] :
( 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)) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2Eschroeder__close__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(ap(c_2Epred__set_2Eschroeder__close(A_27a),V0f),V1s),V2x))
<=> ? [V3n] :
( mem(V3n,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)),V3n),V1s))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESCHROEDER__CLOSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2s] :
( 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] :
( mem(V3n,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)),V3n),V2s))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESCHROEDER__CLOSED,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27a))
=> ! [V1s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2ESCHROEDER__CLOSE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27a))
=> ! [V1s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2ESCHROEDER__CLOSE__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESCHROEDER__BERNSTEIN__AUTO,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))
& ? [V2f] :
( mem(V2f,arr(A_27a,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27a),V2f),V0s),V1t)) ) )
=> ? [V3g] :
( mem(V3g,arr(A_27a,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27a),V3g),V0s),V1t)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESCHROEDER__BERNSTEIN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( ( ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V0s),V1t)) )
& ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27a),V3g),V1t),V0s)) ) )
=> ? [V4h] :
( mem(V4h,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V4h),V0s),V1t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__INJ__SURJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( ( ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V2f),V0s),V1t)) )
& ? [V3g] :
( mem(V3g,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V3g),V0s),V1t)) ) )
=> ? [V4h] :
( mem(V4h,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V4h),V0s),V1t)) ) ) ) ) ) ) ).
fof(lameq_f272,axiom,
! [A_27b,A_27a,V3y] :
( mem(V3y,A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V4x] : ap(f272(A_27b,A_27a,V3y,V0f,V1s),V4x) = 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__ALT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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] :
( mem(V3y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3y),V2t))
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f272(A_27b,A_27a,V3y,V0f,V1s))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__INSERT__IMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1e] :
( mem(V1e,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( 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] :
( 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)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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) ) ) ) ) ) ) ).
fof(lameq_f273,axiom,
! [A_27b,A_27a,V2y] :
( mem(V2y,A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V3x] : ap(f273(A_27b,A_27a,V2y,V0f,V1s),V3x) = 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)) ) ) ) ).
fof(ax_thm_2Epred__set_2ELINV__OPT__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2y] :
( 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),f273(A_27b,A_27a,V2y,V0f,V1s)))),c_2Eoption_2ENONE(A_27a)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ELINV__OPT__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2y] :
( mem(V2y,A_27b)
=> ! [V3x] :
( 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 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__LINV__OPT__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27b,A_27a))
=> ! [V1s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__LINV__OPT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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)) ) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2ELINV__LO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2y] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ELINV__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V0f),V1s),V2t))
=> ! [V3x] :
( 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 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__LINV__INV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t))
=> ! [V3x] :
( 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 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__LINV__BIJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__IFF__INV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V0f),V1s),V2t))
<=> ( ! [V3x] :
( 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] :
( mem(V4g,arr(A_27b,A_27a))
& ! [V5x] :
( 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] :
( mem(V6x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V1s))
=> ap(V4g,ap(V0f,V6x)) = V6x ) )
& ! [V7x] :
( mem(V7x,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V7x),V2t))
=> ap(V0f,ap(V4g,V7x)) = V7x ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1e] :
( mem(V1e,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3t] :
( 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)) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2ERINV__LO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2y] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ERINV__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t))
=> ! [V3x] :
( 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 ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESURJ__INJ__INV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27b,bool))
=> ( p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27a,A_27b),V0f),V1s),V2t))
=> ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EINJ(A_27b,A_27a),V3g),V2t),V1s))
& ! [V4y] :
( mem(V4y,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V4y),V2t))
=> ap(V0f,ap(V3g,V4y)) = V4y ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EFINITE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ! [V1P] :
( mem(V1P,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(V1P,c_2Epred__set_2EEMPTY(A_27a)))
& ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(V1P,V2s))
=> ! [V3e] :
( mem(V3e,A_27a)
=> p(ap(V1P,ap(ap(c_2Epred__set_2EINSERT(A_27a),V3e),V2s))) ) ) ) )
=> p(ap(V1P,V0s)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EEMPTY(A_27a))) ) ).
fof(conj_thm_2Epred__set_2EFINITE__INDUCT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(V0P,c_2Epred__set_2EEMPTY(A_27a)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(V0P,V1s)) )
=> ! [V2e] :
( 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] :
( mem(V3s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3s))
=> p(ap(V0P,V3s)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__REST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__REST__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V1t))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__FINITE__I,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__DIFF__down,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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)))) ) ) ).
fof(conj_thm_2Epred__set_2ESING__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2ESING(A_27a),V0s))
=> p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( 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))) ) ) ) ) ) ).
fof(lameq_f274,axiom,
! [A_27b,A_27a,V1x] :
( mem(V1x,A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2y] : ap(f274(A_27b,A_27a,V1x,V0f),V2y) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2y),ap(ap(c_2Emin_2E_3D(A_27b),V1x),ap(V0f,V2y))) ) ) ).
fof(conj_thm_2Epred__set_2EFINITELY__INJECTIVE__IMAGE__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1x] :
( mem(V1x,A_27b)
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f274(A_27b,A_27a,V1x,V0f)))) )
=> ! [V3s] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJECTIVE__IMAGE__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ap(V0f,V1x) = ap(V0f,V2y)
<=> V1x = V2y ) ) )
=> ! [V3s] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EREL__RESTRICT__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EREL__RESTRICT__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2EREL__RESTRICT__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27a,bool))
=> ! [V2R] :
( 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))) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2ECARD__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ap(c_2Epred__set_2ECARD(A_27a),c_2Epred__set_2EEMPTY(A_27a)) = c_2Enum_2E0
& ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V0s)) = 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Epred__set_2ECARD(A_27a),c_2Epred__set_2EEMPTY(A_27a)) = c_2Enum_2E0 ) ).
fof(conj_thm_2Epred__set_2ECARD__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1x),V0s)) = 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__EQ__0,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = c_2Enum_2E0
<=> V0s = c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V1x)) = 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,c_2Earithmetic_2EZERO)))),ap(c_2Epred__set_2ECARD(A_27a),V0s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__INTER__LESS__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1t))
=> 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))) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Epred__set_2ECARD(A_27a),V1t)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__UNION__EQN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1t)) = 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__PSUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__EQ__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( mem(V1t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1t))
& ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(c_2Epred__set_2ECARD(A_27a),V1t)
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) )
=> V0s = V1t ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EINSERT(A_27a),V0x),c_2Epred__set_2EEMPTY(A_27a))) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ) ) ).
fof(conj_thm_2Epred__set_2ESING__IFF__CARD1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2ESING(A_27a),V0s))
<=> ( ap(c_2Epred__set_2ECARD(A_27a),V0s) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0t))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V0t)) = 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__DIFF__EQN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V0t)) = 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ELESS__CARD__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0t))
=> ! [V1s] :
( 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,c_2Enum_2E0),ap(c_2Epred__set_2ECARD(A_27a),ap(ap(c_2Epred__set_2EDIFF(A_27a),V1s),V0t)))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__FINITE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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] :
( mem(V3N,ty_2Enum_2Enum)
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V3N),V4n))
=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),ap(V0f,V4n)),V2t)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__BIJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))
& ap(c_2Epred__set_2ECARD(A_27a),V1s) = ap(c_2Epred__set_2ECARD(A_27b),V2t) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__BIJ__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) )
=> ap(c_2Epred__set_2ECARD(A_27a),V1s) = ap(c_2Epred__set_2ECARD(A_27b),V2t) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__BIJ__CARD__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0S] :
( mem(V0S,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0S))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2f] :
( 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)) )
=> ap(c_2Epred__set_2ECARD(A_27a),V0S) = ap(c_2Epred__set_2ECARD(A_27b),V1t) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__INJ__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ( ap(V0f,V2x) = ap(V0f,V3y)
<=> V2x = V3y ) ) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1s)) )
=> 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) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESURJ__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ! [V2t] :
( 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))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__SURJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__SURJ__BIJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2t] :
( 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))
& ap(c_2Epred__set_2ECARD(A_27b),V2t) = ap(c_2Epred__set_2ECARD(A_27a),V0s) )
=> p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V1f),V0s),V2t)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__COMPLETE__INDUCTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ! [V1x] :
( mem(V1x,arr(A_27a,bool))
=> ( ! [V2y] :
( 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] :
( mem(V3x,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V3x))
=> p(ap(V0P,V3x)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__CARD__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2s] :
( 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))
=> ap(c_2Epred__set_2ECARD(A_27b),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V0f),V2s)) = ap(c_2Epred__set_2ECARD(A_27a),V2s) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPHP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__CARD__IMAGE__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))
=> 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) ) ) ) ) ) ) ) ).
fof(lameq_f275,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1m] : ap(f275(V0n),V1m) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,bool),V1m),ap(ap(c_2Eprim__rec_2E_3C,V1m),V0n)) ) ).
fof(ax_thm_2Epred__set_2Ecount__def,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Ecount,V0n) = ap(c_2Epred__set_2EGSPEC(ty_2Enum_2Enum,ty_2Enum_2Enum),f275(V0n)) ) ).
fof(conj_thm_2Epred__set_2EIN__COUNT,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V0m),ap(c_2Epred__set_2Ecount,V1n)))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOUNT__applied,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Epred__set_2Ecount,V1n),V0m))
<=> p(ap(ap(c_2Eprim__rec_2E_3C,V0m),V1n)) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOUNT__ZERO,axiom,
ap(c_2Epred__set_2Ecount,c_2Enum_2E0) = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum) ).
fof(conj_thm_2Epred__set_2ECOUNT__SUC,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Ecount,ap(c_2Enum_2ESUC,V0n)) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0n),ap(c_2Epred__set_2Ecount,V0n)) ) ).
fof(conj_thm_2Epred__set_2EFINITE__COUNT,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V0n))) ) ).
fof(conj_thm_2Epred__set_2ECARD__COUNT,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2ECARD(ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V0n)) = V0n ) ).
fof(conj_thm_2Epred__set_2ECOUNT__11,axiom,
! [V0n1] :
( mem(V0n1,ty_2Enum_2Enum)
=> ! [V1n2] :
( mem(V1n2,ty_2Enum_2Enum)
=> ( ap(c_2Epred__set_2Ecount,V0n1) = ap(c_2Epred__set_2Ecount,V1n2)
<=> V0n1 = V1n2 ) ) ) ).
fof(conj_thm_2Epred__set_2ENOT__IN__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a)))
<=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ? [V1x] :
( mem(V1x,A_27a)
& ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINFINITE__INHAB,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ? [V1x] :
( mem(V1x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0P)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__11__INFINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ap(V0f,V1x) = ap(V0f,V2y)
=> V1x = V2y ) ) )
=> ! [V3s] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINFINITE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__INFINITE__NOT__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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] :
( 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_2EINFINITE__INJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINFINITE__INJ__NOT__SURJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ? [V1f] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINFINITE__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a)))
<=> ? [V0f] :
( mem(V0f,arr(A_27a,A_27a))
& ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ap(V0f,V1x) = ap(V0f,V2y)
=> V1x = V2y ) ) )
& ? [V3y] :
( mem(V3y,A_27a)
& ! [V4x] :
( mem(V4x,A_27a)
=> ap(V0f,V4x) != V3y ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINFINITE__NUM__UNIV,axiom,
~ p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ).
fof(conj_thm_2Epred__set_2EFINITE__PSUBSET__INFINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ! [V1t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__PSUBSET__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),c_2Epred__set_2EUNIV(A_27a)))
<=> ! [V0s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINFINITE__DIFF__FINITE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__BIJ__COUNT__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ? [V1c] :
( mem(V1c,arr(ty_2Enum_2Enum,A_27a))
& ? [V2n] :
( mem(V2n,ty_2Enum_2Enum)
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),V1c),ap(c_2Epred__set_2Ecount,V2n)),V0s)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__BIJ__COUNT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ? [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
& ? [V2b] :
( mem(V2b,ty_2Enum_2Enum)
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),V1f),ap(c_2Epred__set_2Ecount,V2b)),V0s)) ) ) ) ) ) ).
fof(lameq_f276,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V4n] : ap(f276(A_27a,V1f,V0s),V4n) = 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))) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__ISO__NUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ? [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,V2n),ap(c_2Epred__set_2ECARD(A_27a),V0s)))
& p(ap(ap(c_2Eprim__rec_2E_3C,V3m),ap(c_2Epred__set_2ECARD(A_27a),V0s))) )
=> ( ap(V1f,V2n) = ap(V1f,V3m)
=> V2n = V3m ) ) ) )
& V0s = ap(c_2Epred__set_2EGSPEC(A_27a,ty_2Enum_2Enum),f276(A_27a,V1f,V0s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__WEAK__ENUMERATE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ? [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
& ? [V2b] :
( mem(V2b,ty_2Enum_2Enum)
& ! [V3e] :
( mem(V3e,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3e),V0s))
<=> ? [V4n] :
( mem(V4n,ty_2Enum_2Enum)
& p(ap(ap(c_2Eprim__rec_2E_3C,V4n),V2b))
& V3e = ap(V1f,V4n) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__WF__noloops,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( 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)))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__StrongOrder__WF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( 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))) ) ) ) ) ).
fof(lameq_f277,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2s] : ap(f277(A_27a,V0P,V1x),V2s) = 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)) ) ) ).
fof(lameq_f278,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1x] : ap(f278(A_27a,V0P),V1x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1x),ap(c_2Ebool_2E_3F(arr(A_27a,bool)),f277(A_27a,V0P,V1x))) ) ).
fof(ax_thm_2Epred__set_2EBIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( 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),f278(A_27a,V0P)) ) ) ).
fof(conj_thm_2Epred__set_2EIN__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1sos] :
( 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] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1sos] :
( mem(V1sos,arr(arr(A_27a,bool),bool))
=> ( p(ap(ap(c_2Epred__set_2EBIGUNION(A_27a),V1sos),V0x))
<=> ? [V2s] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__BIGUNION__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2y] :
( 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] :
( 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))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Epred__set_2EBIGUNION(A_27a),c_2Epred__set_2EEMPTY(arr(A_27a,bool))) = c_2Epred__set_2EEMPTY(A_27a) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__EQ__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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 ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__PAIR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(arr(A_27a,bool),bool))
=> ! [V1s2] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( mem(V0s,arr(arr(A_27a,bool),bool))
=> ! [V1t] :
( 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] :
( 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] :
( mem(V3s,arr(arr(A_27a,bool),bool))
=> ! [V4t] :
( 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] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0X] :
( mem(V0X,arr(A_27a,bool))
=> ! [V1P] :
( 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] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__IMAGE__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ! [V1N] :
( mem(V1N,ty_2Enum_2Enum)
=> ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V1N),V2n))
=> ap(V0f,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,V1N))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V0P))
& ! [V1s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__BIGUNION__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( 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] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__BIGUNION__I,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,arr(A_27a,bool))
=> ! [V1P] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__BIGUNION__SAME__SIZED__SETS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1s] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ! [V2e] :
( 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))
& ap(c_2Epred__set_2ECARD(A_27a),V2e) = V0n ) ) )
& ! [V3e1] :
( mem(V3e1,arr(A_27a,bool))
=> ! [V4e2] :
( 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)) ) ) ) )
=> ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EBIGUNION(A_27a),V1s)) = ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(arr(A_27a,bool)),V1s)),V0n) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__COUNT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(A_27a,bool)))
=> ( ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( V1m != V2n
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0f,V1m)),ap(V0f,V2n))) ) ) )
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),ap(V0f,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,V3n))))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFORALL__IN__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ! [V2x] :
( 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] :
( mem(V3t,arr(A_27a,bool))
=> ! [V4x] :
( 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)) ) ) ) ) ) ) ) ).
fof(lameq_f279,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2s] : ap(f279(A_27a,V0P,V1x),V2s) = 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)) ) ) ).
fof(lameq_f280,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1x] : ap(f280(A_27a,V0P),V1x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1x),ap(c_2Ebool_2E_21(arr(A_27a,bool)),f279(A_27a,V0P,V1x))) ) ).
fof(ax_thm_2Epred__set_2EBIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( 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),f280(A_27a,V0P)) ) ) ).
fof(conj_thm_2Epred__set_2EIN__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1B] :
( 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] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGINTER__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0B] :
( mem(V0B,arr(arr(A_27a,bool),bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(ap(c_2Epred__set_2EBIGINTER(A_27a),V0B),V1x))
<=> ! [V2P] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__BIGINTER__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1f] :
( mem(V1f,arr(A_27b,arr(A_27a,bool)))
=> ! [V2s] :
( 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] :
( 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))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGINTER__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1B] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGINTER__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Epred__set_2EBIGINTER(A_27a),c_2Epred__set_2EEMPTY(arr(A_27a,bool))) = c_2Epred__set_2EUNIV(A_27a) ) ).
fof(conj_thm_2Epred__set_2EBIGINTER__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGINTER__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( 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 ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0X] :
( mem(V0X,arr(A_27a,bool))
=> ! [V1P] :
( 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] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDISJOINT__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0X] :
( mem(V0X,arr(A_27a,bool))
=> ! [V1Y] :
( mem(V1Y,arr(A_27a,bool))
=> ! [V2P] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGINTER__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(arr(A_27a,bool),bool))
=> ! [V1s2] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGINTER__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( ! [V2t] :
( 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)) ) ) ) ) ).
fof(lameq_f281,axiom,
! [A_27a,V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V2u] : ap(f281(A_27a,V0sp),V2u) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V2u) ) ).
fof(conj_thm_2Epred__set_2EDIFF__BIGINTER1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1s] :
( 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)),f281(A_27a,V0sp)),V1s)) ) ) ) ).
fof(lameq_f282,axiom,
! [A_27a,V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V3u] : ap(f282(A_27a,V0sp),V3u) = ap(ap(c_2Epred__set_2EDIFF(A_27a),V0sp),V3u) ) ).
fof(conj_thm_2Epred__set_2EDIFF__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0sp] :
( mem(V0sp,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( ! [V2t] :
( 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)),f282(A_27a,V0sp)),V1s))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ( ? [V1s] :
( 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))) ) ) ) ).
fof(lameq_f283,axiom,
! [A_27a,A_27b,V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V2p] : ap(f283(A_27a,A_27b,V1Q,V0P),V2p) = 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))) ) ) ).
fof(ax_thm_2Epred__set_2ECROSS__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( 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)),f283(A_27a,A_27b,V1Q,V0P)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__CROSS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECROSS__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECROSS__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECROSS__EMPTY__EQN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECROSS__INSERT__LEFT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECROSS__INSERT__RIGHT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__CROSS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECROSS__SINGS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__SING__CROSS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1P] :
( mem(V1P,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V1P))
=> 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)) = ap(c_2Epred__set_2ECARD(A_27b),V1P) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__CROSS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1Q)) )
=> ap(c_2Epred__set_2ECARD(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0P),V1Q)) = ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V0P)),ap(c_2Epred__set_2ECARD(A_27b),V1Q)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECROSS__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,bool))
=> ! [V2P0] :
( mem(V2P0,arr(A_27a,bool))
=> ! [V3Q0] :
( 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)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__CROSS__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECROSS__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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)) ) ) ).
fof(conj_thm_2Epred__set_2EINFINITE__PAIR__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> 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))) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__INL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Esum_2Esum(A_27a,A_27b),bool))
=> ( ! [V2x] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINJ__INR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(ty_2Esum_2Esum(A_27b,A_27a),bool))
=> ( ! [V2x] :
( 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)) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2ECOMPL__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( 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) ) ) ).
fof(conj_thm_2Epred__set_2EIN__COMPL,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_2ECOMPL(A_27a),V1s)))
<=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOMPL__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_2ECOMPL(A_27a),V1s),V0x))
<=> ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V0x),V1s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOMPL__COMPL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(c_2Epred__set_2ECOMPL(A_27a),ap(c_2Epred__set_2ECOMPL(A_27a),V0s)) = V0s ) ) ).
fof(conj_thm_2Epred__set_2ECOMPL__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOMPL__SPLITS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1q] :
( 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 ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__UNION__COMPL,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_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))) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOMPL__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Epred__set_2ECOMPL(A_27a),c_2Epred__set_2EEMPTY(A_27a)) = c_2Epred__set_2EUNIV(A_27a) ) ).
fof(conj_thm_2Epred__set_2ECOMPL__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOMPL__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2EITSET__ind,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1P] :
( mem(V1P,arr(arr(A_27a,bool),arr(A_27b,bool)))
=> ( ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3b] :
( 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] :
( mem(V4v,arr(A_27a,bool))
=> ! [V5v1] :
( mem(V5v1,A_27b)
=> p(ap(ap(V1P,V4v),V5v1)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EITSET__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V2b] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EITSET__IND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1P] :
( mem(V1P,arr(arr(A_27a,bool),arr(A_27b,bool)))
=> ( ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3b] :
( 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] :
( mem(V4v,arr(A_27a,bool))
=> ! [V5v1] :
( mem(V5v1,A_27b)
=> p(ap(ap(V1P,V4v),V5v1)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EITSET__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V2b] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EITSET__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ap(ap(ap(c_2Epred__set_2EITSET(A_27a,A_27b),V0f),c_2Epred__set_2EEMPTY(A_27a)),V1b) = V1b ) ) ) ) ).
fof(conj_thm_2Epred__set_2EITSET__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3b] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOMMUTING__ITSET__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4z] :
( 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] :
( mem(V5x,A_27a)
=> ! [V6b] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOMMUTING__ITSET__RECURSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1e] :
( mem(V1e,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ! [V3b] :
( mem(V3b,A_27b)
=> ( ( ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ! [V6z] :
( 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)) ) ) ) ) ) ) ) ).
fof(lameq_f284,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2e] :
( mem(V2e,A_27a)
=> ! [V3acc] : ap(f284(A_27a,V0f,V2e),V3acc) = ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V2e)),V3acc) ) ) ).
fof(lameq_f285,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2e] : ap(f285(A_27a,V0f),V2e) = f284(A_27a,V0f,V2e) ) ).
fof(ax_thm_2Epred__set_2ESUM__IMAGE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s) = ap(ap(ap(c_2Epred__set_2EITSET(A_27a,ty_2Enum_2Enum),f285(A_27a,V0f)),V1s),c_2Enum_2E0) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),c_2Epred__set_2EEMPTY(A_27a)) = c_2Enum_2E0
& ! [V1e] :
( mem(V1e,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)) = 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1e] :
( mem(V1e,A_27a)
=> 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))) = ap(V0f,V1e) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__SUBSET__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__IN__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2e] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ! [V2e] :
( mem(V2e,A_27a)
=> ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EDELETE(A_27a),V1s),V2e)) = 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V2t)) )
=> ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EUNION(A_27a),V1s),V2t)) = 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__lower__bound,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,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)),V2n)),ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__upper__bound,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ( ! [V3x] :
( 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)),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)),V2n))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__SAME__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0P))
=> ! [V1f] :
( mem(V1f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2p] :
( mem(V2p,A_27a)
=> ( ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2p),V0P))
& ! [V3q] :
( mem(V3q,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3q),V0P))
=> ap(V1f,V2p) = ap(V1f,V3q) ) ) )
=> ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V1f),V0P) = ap(ap(c_2Earithmetic_2E_2A,ap(c_2Epred__set_2ECARD(A_27a),V0P)),ap(V1f,V2p)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27a,bool))
=> ! [V2f1] :
( mem(V2f1,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3f2] :
( mem(V3f2,arr(A_27a,ty_2Enum_2Enum))
=> ( ( V0s1 = V1s2
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1s2))
=> ap(V2f1,V4x) = ap(V3f2,V4x) ) ) )
=> ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V2f1),V0s1) = ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V3f2),V1s2) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__ZERO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V0f),V1s) = c_2Enum_2E0
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ap(V0f,V2x) = c_2Enum_2E0 ) ) ) ) ) ) ) ).
fof(lameq_f286,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3x] : ap(f286(A_27a,V0f,V1g),V3x) = ap(ap(c_2Earithmetic_2EABS__DIFF,ap(V0f,V3x)),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Epred__set_2EABS__DIFF__SUM__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( 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),f286(A_27a,V0f,V1g)),V2s))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__MONO__LESS__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ( ! [V3x] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__MONO__LESS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1g] :
( mem(V1g,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ( ( ? [V3x] :
( 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] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__INJ__o,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27b,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27b),V0s))
=> ! [V1g] :
( 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] :
( mem(V2f,arr(A_27a,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)) = ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27b),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Enum_2Enum,A_27a),V2f),V1g)),V0s) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__IMAGE__PERMUTES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27a))
=> ( p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27a),V1g),V0s),V0s))
=> ! [V2f] :
( mem(V2f,arr(A_27a,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) = ap(ap(c_2Epred__set_2ESUM__IMAGE(A_27a),V2f),V0s) ) ) ) ) ) ) ).
fof(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)) ).
fof(conj_thm_2Epred__set_2ESUM__SET__THM,axiom,
( ap(c_2Epred__set_2ESUM__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = c_2Enum_2E0
& ! [V0x] :
( mem(V0x,ty_2Enum_2Enum)
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s))
=> ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0x),V1s)) = ap(ap(c_2Earithmetic_2E_2B,V0x),ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EDELETE(ty_2Enum_2Enum),V1s),V0x))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__SET__EMPTY,axiom,
ap(c_2Epred__set_2ESUM__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = c_2Enum_2E0 ).
fof(conj_thm_2Epred__set_2ESUM__SET__SING,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0n),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))) = V0n ) ).
fof(conj_thm_2Epred__set_2ESUM__SET__SUBSET__LE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ! [V1t] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__SET__IN__LE,axiom,
! [V0x] :
( mem(V0x,ty_2Enum_2Enum)
=> ! [V1s] :
( 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),V0x),V1s)) )
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V0x),ap(c_2Epred__set_2ESUM__SET,V1s))) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__SET__DELETE,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
=> ! [V1e] :
( mem(V1e,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EDELETE(ty_2Enum_2Enum),V0s),V1e)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Enum_2Enum),ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1e),V0s)),ap(ap(c_2Earithmetic_2E_2D,ap(c_2Epred__set_2ESUM__SET,V0s)),V1e)),ap(c_2Epred__set_2ESUM__SET,V0s)) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUM__SET__UNION,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ! [V1t] :
( 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)) )
=> ap(c_2Epred__set_2ESUM__SET,ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V0s),V1t)) = 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))) ) ) ) ).
fof(lameq_f287,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2e] :
( mem(V2e,A_27a)
=> ! [V3acc] : ap(f287(A_27a,V0f,V2e),V3acc) = ap(ap(c_2Earithmetic_2E_2A,ap(V0f,V2e)),V3acc) ) ) ).
fof(lameq_f288,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V2e] : ap(f288(A_27a,V0f),V2e) = f287(A_27a,V0f,V2e) ) ).
fof(ax_thm_2Epred__set_2EPROD__IMAGE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2EPROD__IMAGE(A_27a),V0f),V1s) = ap(ap(ap(c_2Epred__set_2EITSET(A_27a,ty_2Enum_2Enum),f288(A_27a,V0f)),V1s),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ).
fof(conj_thm_2Epred__set_2EPROD__IMAGE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ( ap(ap(c_2Epred__set_2EPROD__IMAGE(A_27a),V0f),c_2Epred__set_2EEMPTY(A_27a)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))
& ! [V1e] :
( mem(V1e,A_27a)
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V2s))
=> ap(ap(c_2Epred__set_2EPROD__IMAGE(A_27a),V0f),ap(ap(c_2Epred__set_2EINSERT(A_27a),V1e),V2s)) = 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))) ) ) ) ) ) ) ).
fof(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)) ).
fof(conj_thm_2Epred__set_2EPROD__SET__THM,axiom,
( ap(c_2Epred__set_2EPROD__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))
& ! [V0x] :
( mem(V0x,ty_2Enum_2Enum)
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s))
=> ap(c_2Epred__set_2EPROD__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0x),V1s)) = ap(ap(c_2Earithmetic_2E_2A,V0x),ap(c_2Epred__set_2EPROD__SET,ap(ap(c_2Epred__set_2EDELETE(ty_2Enum_2Enum),V1s),V0x))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPROD__SET__EMPTY,axiom,
ap(c_2Epred__set_2EPROD__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)) ).
fof(conj_thm_2Epred__set_2EPROD__SET__IMAGE__REDUCTION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( 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))) )
=> 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))) = 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))) ) ) ) ) ) ).
fof(ax_thm_2Epred__set_2EMAX__SET__DEF,axiom,
! [V0s] :
( 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] :
( mem(V1y,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1y),V0s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V1y),ap(c_2Epred__set_2EMAX__SET,V0s))) ) ) ) )
& ( V0s = c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)
=> ap(c_2Epred__set_2EMAX__SET,V0s) = c_2Enum_2E0 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EMAX__SET__THM,axiom,
( ap(c_2Epred__set_2EMAX__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = c_2Enum_2E0
& ! [V0e] :
( mem(V0e,ty_2Enum_2Enum)
=> ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s))
=> ap(c_2Epred__set_2EMAX__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0e),V1s)) = ap(ap(c_2Earithmetic_2EMAX,V0e),ap(c_2Epred__set_2EMAX__SET,V1s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EMAX__SET__REWRITES,axiom,
! [V0e] :
( mem(V0e,ty_2Enum_2Enum)
=> ( ap(c_2Epred__set_2EMAX__SET,c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)) = c_2Enum_2E0
& ap(c_2Epred__set_2EMAX__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0e),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))) = V0e ) ) ).
fof(conj_thm_2Epred__set_2EMAX__SET__ELIM,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ! [V1Q] :
( 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,c_2Enum_2E0)) )
& ! [V2x] :
( mem(V2x,ty_2Enum_2Enum)
=> ( ( ! [V3y] :
( mem(V3y,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3y),V0P))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V3y),V2x)) ) )
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2x),V0P)) )
=> p(ap(V1Q,V2x)) ) ) )
=> p(ap(V1Q,ap(c_2Epred__set_2EMAX__SET,V0P))) ) ) ) ).
fof(ax_thm_2Epred__set_2EMIN__SET__DEF,axiom,
c_2Epred__set_2EMIN__SET = c_2Ewhile_2ELEAST ).
fof(conj_thm_2Epred__set_2EMIN__SET__ELIM,axiom,
! [V0P] :
( mem(V0P,arr(ty_2Enum_2Enum,bool))
=> ! [V1Q] :
( mem(V1Q,arr(ty_2Enum_2Enum,bool))
=> ( ( V0P != c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)
& ! [V2x] :
( mem(V2x,ty_2Enum_2Enum)
=> ( ( ! [V3y] :
( mem(V3y,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V3y),V0P))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V2x),V3y)) ) )
& p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V2x),V0P)) )
=> p(ap(V1Q,V2x)) ) ) )
=> p(ap(V1Q,ap(c_2Epred__set_2EMIN__SET,V0P))) ) ) ) ).
fof(conj_thm_2Epred__set_2EMIN__SET__THM,axiom,
( ! [V0e] :
( mem(V0e,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2EMIN__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0e),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum))) = V0e )
& ! [V1s] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ! [V2e1] :
( mem(V2e1,ty_2Enum_2Enum)
=> ! [V3e2] :
( mem(V3e2,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2EMIN__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V2e1),ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V3e2),V1s))) = ap(ap(c_2Earithmetic_2EMIN,V2e1),ap(c_2Epred__set_2EMIN__SET,ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V3e2),V1s))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EMIN__SET__LEM,axiom,
! [V0s] :
( 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] :
( mem(V1x,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1x),V0s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,ap(c_2Epred__set_2EMIN__SET,V0s)),V1x)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__MIN__SET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0I] :
( mem(V0I,arr(ty_2Enum_2Enum,bool))
=> ! [V1J] :
( mem(V1J,arr(ty_2Enum_2Enum,bool))
=> ! [V2n] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__MAX__SET,axiom,
! [V0I] :
( mem(V0I,arr(ty_2Enum_2Enum,bool))
=> ! [V1J] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2EMIN__SET__LEQ__MAX__SET,axiom,
! [V0s] :
( 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))) ) ) ).
fof(conj_thm_2Epred__set_2EMIN__SET__UNION,axiom,
! [V0A] :
( mem(V0A,arr(ty_2Enum_2Enum,bool))
=> ! [V1B] :
( 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) )
=> ap(c_2Epred__set_2EMIN__SET,ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V0A),V1B)) = ap(ap(c_2Earithmetic_2EMIN,ap(c_2Epred__set_2EMIN__SET,V0A)),ap(c_2Epred__set_2EMIN__SET,V1B)) ) ) ) ).
fof(conj_thm_2Epred__set_2EMAX__SET__UNION,axiom,
! [V0A] :
( mem(V0A,arr(ty_2Enum_2Enum,bool))
=> ! [V1B] :
( 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)) )
=> ap(c_2Epred__set_2EMAX__SET,ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),V0A),V1B)) = ap(ap(c_2Earithmetic_2EMAX,ap(c_2Epred__set_2EMAX__SET,V0A)),ap(c_2Epred__set_2EMAX__SET,V1B)) ) ) ) ).
fof(lameq_f289,axiom,
! [A_27a,V0set] :
( mem(V0set,arr(A_27a,bool))
=> ! [V1s] : ap(f289(A_27a,V0set),V1s) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V1s),ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V0set)) ) ).
fof(ax_thm_2Epred__set_2EPOW__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0set] :
( 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)),f289(A_27a,V0set)) ) ) ).
fof(conj_thm_2Epred__set_2EIN__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0set] :
( mem(V0set,arr(A_27a,bool))
=> ! [V1e] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNIV__FUN__TO__BOOL,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Epred__set_2EUNIV(arr(A_27a,bool)) = ap(c_2Epred__set_2EPOW(A_27a),c_2Epred__set_2EUNIV(A_27a)) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__INSERT__RIGHT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__DELETE__BOTH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27a,bool))
=> ! [V2x] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPOW__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(c_2Epred__set_2EPOW(A_27a),V0s) != c_2Epred__set_2EEMPTY(arr(A_27a,bool)) ) ) ).
fof(conj_thm_2Epred__set_2EPOW__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0e] :
( mem(V0e,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ).
fof(lameq_f290,axiom,
! [A_27a,V0e] :
( mem(V0e,A_27a)
=> ! [V2ps] : ap(f290(A_27a,V0e),V2ps) = 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) ) ).
fof(conj_thm_2Epred__set_2EPOW__EQNS,axiom,
! [A_27a] :
( ne(A_27a)
=> ( 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] :
( mem(V0e,A_27a)
=> ! [V1s] :
( 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)),f290(A_27a,V0e)),ap(c_2Epred__set_2EPOW(A_27a),V1s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__POW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ap(c_2Epred__set_2ECARD(arr(A_27a,bool)),ap(c_2Epred__set_2EPOW(A_27a),V0s)) = ap(ap(c_2Earithmetic_2EEXP,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO))),ap(c_2Epred__set_2ECARD(A_27a),V0s)) ) ) ) ).
fof(lameq_f291,axiom,
! [A_27a,V0x] : ap(f291(A_27a),V0x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V0x),c_2Ebool_2EF) ).
fof(conj_thm_2Epred__set_2EGSPEC__F,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f291(A_27a)) = c_2Epred__set_2EEMPTY(A_27a) ) ).
fof(lameq_f292,axiom,
! [A_27a,V0x] : ap(f292(A_27a),V0x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V0x),c_2Ebool_2ET) ).
fof(conj_thm_2Epred__set_2EGSPEC__T,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f292(A_27a)) = c_2Epred__set_2EUNIV(A_27a) ) ).
fof(lameq_f293,axiom,
! [A_27a,V0y] :
( mem(V0y,arr(A_27a,bool))
=> ! [V1x] : ap(f293(A_27a,V0y),V1x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1x),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0y)) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0y] :
( mem(V0y,arr(A_27a,bool))
=> ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f293(A_27a,V0y)) = V0y ) ) ).
fof(lameq_f294,axiom,
! [A_27a,V0y] :
( mem(V0y,A_27a)
=> ! [V1x] : ap(f294(A_27a,V0y),V1x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1x),ap(ap(c_2Emin_2E_3D(A_27a),V1x),V0y)) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0y] :
( mem(V0y,A_27a)
=> ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f294(A_27a,V0y)) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0y),c_2Epred__set_2EEMPTY(A_27a)) ) ) ).
fof(lameq_f295,axiom,
! [A_27a,V0y] :
( mem(V0y,A_27a)
=> ! [V1x] : ap(f295(A_27a,V0y),V1x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V1x),ap(ap(c_2Emin_2E_3D(A_27a),V0y),V1x)) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__EQ2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0y] :
( mem(V0y,A_27a)
=> ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f295(A_27a,V0y)) = ap(ap(c_2Epred__set_2EINSERT(A_27a),V0y),c_2Epred__set_2EEMPTY(A_27a)) ) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__F__COND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Epair_2Eprod(A_27b,bool)))
=> ( ! [V1x] :
( 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) ) ) ) ) ).
fof(lameq_f296,axiom,
! [A_27a,V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V2x] : ap(f296(A_27a,V1Q,V0P),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V2x)),ap(V1Q,V2x))) ) ) ).
fof(lameq_f297,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V3x] : ap(f297(A_27a,V0P),V3x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V3x),ap(V0P,V3x)) ) ).
fof(lameq_f298,axiom,
! [A_27a,V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V4x] : ap(f298(A_27a,V1Q),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(V1Q,V4x)) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__AND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f296(A_27a,V1Q,V0P)) = ap(ap(c_2Epred__set_2EINTER(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f297(A_27a,V0P))),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f298(A_27a,V1Q))) ) ) ) ).
fof(lameq_f299,axiom,
! [A_27a,V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V2x] : ap(f299(A_27a,V1Q,V0P),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(ap(c_2Ebool_2E_5C_2F,ap(V0P,V2x)),ap(V1Q,V2x))) ) ) ).
fof(conj_thm_2Epred__set_2EGSPEC__OR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f299(A_27a,V1Q,V0P)) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f297(A_27a,V0P))),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f298(A_27a,V1Q))) ) ) ) ).
fof(ax_thm_2Epred__set_2Eequiv__on__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s))
<=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> p(ap(ap(V0R,V2x),V2x)) ) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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] :
( mem(V5x,A_27a)
=> ! [V6y] :
( mem(V6y,A_27a)
=> ! [V7z] :
( 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)) ) ) ) ) ) ) ) ) ) ).
fof(lameq_f300,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V4y] : ap(f300(A_27a,V1s,V3x,V0R),V4y) = 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))) ) ) ) ).
fof(lameq_f301,axiom,
! [A_27a,V2t] :
( mem(V2t,arr(A_27a,bool))
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V3x] : ap(f301(A_27a,V2t,V0R,V1s),V3x) = 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),f300(A_27a,V1s,V3x,V0R)))) ) ) ) ).
fof(lameq_f302,axiom,
! [A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] : ap(f302(A_27a,V0R,V1s),V2t) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V2t),ap(c_2Ebool_2E_3F(A_27a),f301(A_27a,V2t,V0R,V1s))) ) ) ).
fof(ax_thm_2Epred__set_2Epartition__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( 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)),f302(A_27a,V0R,V1s)) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIGUNION__partition,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EEMPTY__NOT__IN__partition,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Epartition__elements__disjoint,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s))
=> ! [V2t1] :
( mem(V2t1,arr(A_27a,bool))
=> ! [V3t2] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Epartition__elements__interrelate,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2Eequiv__on(A_27a),V0R),V1s))
=> ! [V2t] :
( 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] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Epartition__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__partition,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( 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] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Epartition__CARD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( 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)) )
=> ap(c_2Epred__set_2ECARD(A_27a),V1s) = 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)) ) ) ) ) ).
fof(ax_thm_2Epred__set_2Epairwise__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2Epairwise(A_27a),V0P),V1s))
<=> ! [V2e1] :
( mem(V2e1,A_27a)
=> ! [V3e2] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Epairwise__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2] :
( 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] :
( mem(V3x,A_27a)
=> ! [V4y] :
( 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)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Epairwise__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(lameq_f303,axiom,
! [A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] : ap(f303(A_27a,V0R,V1x),V2y) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2y),ap(ap(V0R,V1x),V2y)) ) ) ).
fof(lameq_f304,axiom,
! [A_27a,V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] : ap(f304(A_27a,V0R,V3x),V4y) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4y),ap(ap(ap(c_2Erelation_2ERTC(A_27a),V0R),V3x),V4y)) ) ) ).
fof(conj_thm_2Epred__set_2EKoenigsLemma,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f303(A_27a,V0R,V1x)))) )
=> ! [V3x] :
( mem(V3x,A_27a)
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f304(A_27a,V0R,V3x))))
=> ? [V5f] :
( mem(V5f,arr(ty_2Enum_2Enum,A_27a))
& ap(V5f,c_2Enum_2E0) = V3x
& ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> p(ap(ap(V0R,ap(V5f,V6n)),ap(V5f,ap(c_2Enum_2ESUC,V6n)))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EKoenigsLemma__WF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( ( ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f303(A_27a,V0R,V1x)))) )
& p(ap(c_2Erelation_2EWF(A_27a),ap(c_2Erelation_2Einv(A_27a,A_27a),V0R))) )
=> ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(c_2Epred__set_2EFINITE(A_27a),ap(c_2Epred__set_2EGSPEC(A_27a,A_27a),f304(A_27a,V0R,V3x)))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESET__EQ__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__EQN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__SUBSET__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_2ESUBSET(A_27a),V1t),V2u)) )
=> p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V2u)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__PSUBSET__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_2EPSUBSET(A_27a),V1t),V2u)) )
=> p(ap(ap(c_2Epred__set_2EPSUBSET(A_27a),V0s),V2u)) ) ) ) ) ) ).
fof(lameq_f305,axiom,
! [A_27a,A_27b,V0a] :
( mem(V0a,A_27a)
=> ! [V3y] : ap(f305(A_27a,A_27b,V0a),V3y) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0a),V3y) ) ).
fof(conj_thm_2Epred__set_2ECROSS__EQNS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1s1] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2] :
( 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)),f305(A_27a,A_27b,V0a)),V2s2)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V1s1),V2s2)) ) ) ) ) ) ) ).
fof(lameq_f306,axiom,
! [V1p] : ap(f306,V1p) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V1p),ap(c_2Epred__set_2Ecount,V1p)) ).
fof(conj_thm_2Epred__set_2Ecount__EQN,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Ecount,V0n) = ap(ap(ap(c_2Ebool_2ECOND(arr(ty_2Enum_2Enum,bool)),ap(ap(c_2Emin_2E_3D(ty_2Enum_2Enum),V0n),c_2Enum_2E0)),c_2Epred__set_2EEMPTY(ty_2Enum_2Enum)),ap(ap(c_2Ebool_2ELET(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,bool)),f306),ap(c_2Eprim__rec_2EPRE,V0n))) ) ).
fof(conj_thm_2Epred__set_2EUNIQUE__MEMBER__SING,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))
& ! [V2y] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Einj__surj,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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] :
( mem(V3f_27,arr(A_27b,A_27a))
& p(ap(ap(ap(c_2Epred__set_2ESURJ(A_27b,A_27a),V3f_27),V2t),V1s)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Einfinite__rest,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(ax_thm_2Epred__set_2Echooser__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2Echooser(A_27a),V0s),c_2Enum_2E0) = ap(c_2Epred__set_2ECHOICE(A_27a),V0s) )
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Epred__set_2Echooser(A_27a),V1s),ap(c_2Enum_2ESUC,V2n)) = ap(ap(c_2Epred__set_2Echooser(A_27a),ap(c_2Epred__set_2EREST(A_27a),V1s)),V2n) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Echooser__def__compute,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ap(ap(c_2Epred__set_2Echooser(A_27a),V0s),c_2Enum_2E0) = ap(c_2Epred__set_2ECHOICE(A_27a),V0s) )
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> ap(ap(c_2Epred__set_2Echooser(A_27a),V1s),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,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,V2n))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) ) )
& ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> ap(ap(c_2Epred__set_2Echooser(A_27a),V3s),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,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,V4n))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Einfinite__num__inj,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ~ p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
<=> ? [V1f] :
( 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)) ) ) ) ) ).
fof(ax_thm_2Epred__set_2Ecountable__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s))
<=> ? [V1f] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Ecountable__image__nats,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( 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)))) ) ) ).
fof(conj_thm_2Epred__set_2Ecountable__surj,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s))
<=> ( V0s = c_2Epred__set_2EEMPTY(A_27a)
| ? [V1f] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Enum__countable,axiom,
p(ap(c_2Epred__set_2Ecountable(ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ).
fof(conj_thm_2Epred__set_2Esubset__countable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Eimage__countable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOUNTABLE__ALT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0s))
<=> ? [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,A_27a))
& ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
=> ? [V3n] :
( mem(V3n,ty_2Enum_2Enum)
& ap(V1f,V3n) = V2x ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOUNTABLE__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),V0s),V1t))
& p(ap(c_2Epred__set_2Ecountable(A_27a),V1t)) )
=> p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Efinite__countable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> p(ap(c_2Epred__set_2Ecountable(A_27a),V0s)) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOUNTABLE__COUNT,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> p(ap(c_2Epred__set_2Ecountable(ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V0n))) ) ).
fof(conj_thm_2Epred__set_2ECOUNTABLE__NUM,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> p(ap(c_2Epred__set_2Ecountable(ty_2Enum_2Enum),V0s)) ) ).
fof(conj_thm_2Epred__set_2ECOUNTABLE__IMAGE__NUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27a))
=> ! [V1s] :
( 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))) ) ) ) ).
fof(ax_thm_2Epred__set_2Enum__to__pair__def,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Enum__to__pair,V0n) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(c_2Enumpair_2Enfst,V0n)),ap(c_2Enumpair_2Ensnd,V0n)) ) ).
fof(ax_thm_2Epred__set_2Epair__to__num__def,axiom,
! [V0m] :
( mem(V0m,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Epair__to__num,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V0m),V1n)) = ap(ap(c_2Enumpair_2Enpair,V0m),V1n) ) ) ).
fof(conj_thm_2Epred__set_2Epair__to__num__formula,axiom,
! [V0x] :
( mem(V0x,ty_2Enum_2Enum)
=> ! [V1y] :
( mem(V1y,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Epair__to__num,ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V0x),V1y)) = 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,V0x),V1y)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))),ap(ap(c_2Earithmetic_2E_2B,V0x),V1y))),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,c_2Earithmetic_2EZERO)))),V1y) ) ) ).
fof(conj_thm_2Epred__set_2Epair__to__num__inv,axiom,
( ! [V0x] :
( mem(V0x,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Epair__to__num,ap(c_2Epred__set_2Enum__to__pair,V0x)) = V0x )
& ! [V1x] :
( mem(V1x,ty_2Enum_2Enum)
=> ! [V2y] :
( mem(V2y,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),V1x),V2y))) = ap(ap(c_2Epair_2E_2C(ty_2Enum_2Enum,ty_2Enum_2Enum),V1x),V2y) ) ) ) ).
fof(conj_thm_2Epred__set_2Ecross__countable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Einter__countable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Einj__countable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Ebigunion__countable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2Ecountable(arr(A_27a,bool)),V0s))
& ! [V1x] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2Eunion__countable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Eunion__countable__IFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Einj__image__countable__IFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Epow__no__surj,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ~ ? [V1f] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2Einfinite__pow__uncountable,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(conj_thm_2Epred__set_2Ecountable__Usum,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Ecountable__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> p(ap(c_2Epred__set_2Ecountable(A_27a),c_2Epred__set_2EEMPTY(A_27a))) ) ).
fof(conj_thm_2Epred__set_2Ecountable__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Ecross__countable__IFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Ecountable__Uprod,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EEXPLICIT__ENUMERATE__MONO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1s] :
( 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)),V0n),V1s)),V1s)) ) ) ) ).
fof(conj_thm_2Epred__set_2EEXPLICIT__ENUMERATE__NOT__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1s] :
( 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)),V0n),V1s) != c_2Epred__set_2EEMPTY(A_27a) ) ) ) ) ).
fof(lameq_f307,axiom,
! [A_27a,V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1n] : ap(f307(A_27a,V0s),V1n) = 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)) ) ).
fof(conj_thm_2Epred__set_2EINFINITE__EXPLICIT__ENUMERATE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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),f307(A_27a,V0s)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s)) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__NUM__COUNTABLE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ? [V1f] :
( 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)) ) ) ) ).
fof(lameq_f308,axiom,
! [A_27a,V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f] : ap(f308(A_27a,V0s),V1f) = ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,A_27a),V1f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),V0s) ) ).
fof(ax_thm_2Epred__set_2Eenumerate__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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)),f308(A_27a,V0s)) ) ) ).
fof(conj_thm_2Epred__set_2EENUMERATE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ? [V1f] :
( 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)) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOUNTABLE__ALT__BIJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECOUNTABLE__ENUM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0c] :
( mem(V0c,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2Ecountable(A_27a),V0c))
<=> ( V0c = c_2Epred__set_2EEMPTY(A_27a)
| ? [V1f] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1M] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( mem(V1s2,arr(A_27a,bool))
=> ! [V2s3] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__SUBSET__EQN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPSUBSET__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,A_27a)
=> ! [V1x] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINTER__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( 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 ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__DELETE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(A_27a,bool))
=> ! [V1B] :
( mem(V1B,arr(A_27a,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDELETE__SUBSET__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1e] :
( mem(V1e,A_27a)
=> ! [V2s2] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__INSERT__EXPAND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2P] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s1] :
( mem(V0s1,arr(A_27a,bool))
=> ! [V1s2] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EINSERT__EQ__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( 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)))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__UNION__LE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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)))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__SUBSET__gen,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2u] :
( mem(V2u,arr(A_27a,bool))
=> ! [V3t] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2ECARD__REST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& V0s != c_2Epred__set_2EEMPTY(A_27a) )
=> ap(c_2Epred__set_2ECARD(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s)) = ap(ap(c_2Earithmetic_2E_2D,ap(c_2Epred__set_2ECARD(A_27a),V0s)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ).
fof(conj_thm_2Epred__set_2ESUBSET__DIFF__EMPTY,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) = c_2Epred__set_2EEMPTY(A_27a)
<=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EDIFF__INTER__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,arr(A_27a,bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EUNION__DIFF__2,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),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V1t)) = V0s ) ) ) ).
fof(conj_thm_2Epred__set_2Ecount__add,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Ecount,ap(ap(c_2Earithmetic_2E_2B,V0n),V1m)) = ap(ap(c_2Epred__set_2EUNION(ty_2Enum_2Enum),ap(c_2Epred__set_2Ecount,V0n)),ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Enum_2Enum),ap(c_2Earithmetic_2E_2B,V0n)),ap(c_2Epred__set_2Ecount,V1m))) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__EQ__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27b,A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2z] :
( 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] :
( mem(V3x,A_27b)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27b),V3x),V1s))
=> ap(V0f,V3x) = V2z ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Ecount__add1,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ap(c_2Epred__set_2Ecount,ap(ap(c_2Earithmetic_2E_2B,V0n),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))) = ap(ap(c_2Epred__set_2EINSERT(ty_2Enum_2Enum),V0n),ap(c_2Epred__set_2Ecount,V0n)) ) ).
fof(conj_thm_2Epred__set_2Ecompl__insert,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2Ein__max__set,axiom,
! [V0s] :
( mem(V0s,arr(ty_2Enum_2Enum,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V0s))
=> ! [V1x] :
( mem(V1x,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),V1x),V0s))
=> p(ap(ap(c_2Earithmetic_2E_3C_3D,V1x),ap(c_2Epred__set_2EMAX__SET,V0s))) ) ) ) ) ).
fof(lameq_f309,axiom,
! [A_27b,A_27a,V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2x] : ap(f309(A_27b,A_27a,V1s,V0f),V2x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V2x),ap(ap(c_2Ebool_2EIN(A_27b),ap(V0f,V2x)),V1s)) ) ) ).
fof(ax_thm_2Epred__set_2EPREIMAGE__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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),f309(A_27b,A_27a,V1s,V0f)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__ALT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIN__PREIMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2x] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__UNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( 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) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__COMPL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__COMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27c))
=> ! [V2s] :
( 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) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__I,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Epred__set_2EPREIMAGE(A_27a,A_27a),c_2Ecombin_2EI(A_27a)) = c_2Ecombin_2EI(arr(A_27a,bool)) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__K,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__DISJOINT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( mem(V1s,arr(A_27b,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__CROSS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)))
=> ! [V1a] :
( mem(V1a,arr(A_27b,bool))
=> ! [V2b] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__COMPL__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ! [V2sp] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EPREIMAGE__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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)))) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EIMAGE__PREIMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1s] :
( 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)) ) ) ) ) ).
fof(ax_thm_2Epred__set_2Eis__measure__maximal__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( 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] :
( 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))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EFINITE__is__measure__maximal,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& V1s != c_2Epred__set_2EEMPTY(A_27a) )
=> ? [V2x] :
( mem(V2x,A_27a)
& p(ap(ap(ap(c_2Epred__set_2Eis__measure__maximal(A_27a),V0m),V1s),V2x)) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Eis__measure__maximal__SING,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0m] :
( mem(V0m,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( 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 ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2Eis__measure__maximal__INSERT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2m] :
( mem(V2m,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3e] :
( mem(V3e,A_27a)
=> ! [V4y] :
( 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)) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------