ITP001 Axioms: ITP003+5.ax
%------------------------------------------------------------------------------
% File : ITP003+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 : bool+2.ax [Gau20]
% : HL4003+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 284 ( 30 unt; 0 def)
% Number of atoms : 1501 ( 171 equ)
% Maximal formula atoms : 21 ( 5 avg)
% Number of connectives : 1263 ( 46 ~; 61 |; 182 &)
% ( 151 <=>; 823 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 102 ( 102 usr; 13 con; 0-5 aty)
% Number of variables : 884 ( 828 !; 56 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Ebool_2Eitself,axiom,
! [A0] :
( ne(A0)
=> ne(ty_2Ebool_2Eitself(A0)) ) ).
fof(mem_c_2Ebool_2E_21,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(ax_all_p,axiom,
! [A] :
( ne(A)
=> ! [Q] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_21(A),Q))
<=> ! [X] :
( mem(X,A)
=> p(ap(Q,X)) ) ) ) ) ).
fof(mem_c_2Ebool_2E_2F_5C,axiom,
mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).
fof(ax_and_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_2F_5C,Q),R))
<=> ( p(Q)
& p(R) ) ) ) ) ).
fof(mem_c_2Ebool_2E_3F,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2E_3F(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(ax_ex_p,axiom,
! [A] :
( ne(A)
=> ! [Q] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_3F(A),Q))
<=> ? [X] :
( mem(X,A)
& p(ap(Q,X)) ) ) ) ) ).
fof(mem_c_2Ebool_2E_3F_21,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2E_3F_21(A_27a),arr(arr(A_27a,bool),bool)) ) ).
fof(mem_c_2Ebool_2EARB,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2EARB(A_27a),A_27a) ) ).
fof(mem_c_2Ebool_2EBOUNDED,axiom,
mem(c_2Ebool_2EBOUNDED,arr(bool,bool)) ).
fof(mem_c_2Ebool_2ECOND,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2ECOND(A_27a),arr(bool,arr(A_27a,arr(A_27a,A_27a)))) ) ).
fof(mem_c_2Ebool_2EDATATYPE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2EDATATYPE(A_27a),arr(A_27a,bool)) ) ).
fof(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
fof(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
fof(mem_c_2Ebool_2EIN,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2EIN(A_27a),arr(A_27a,arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Ebool_2ELET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ebool_2ELET(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(A_27a,A_27b))) ) ) ).
fof(mem_c_2Ebool_2EONE__ONE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ebool_2EONE__ONE(A_27a,A_27b),arr(arr(A_27a,A_27b),bool)) ) ) ).
fof(mem_c_2Ebool_2EONTO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ebool_2EONTO(A_27a,A_27b),arr(arr(A_27a,A_27b),bool)) ) ) ).
fof(mem_c_2Ebool_2ERES__ABSTRACT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27a,A_27b),arr(A_27a,A_27b)))) ) ) ).
fof(mem_c_2Ebool_2ERES__EXISTS,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2ERES__EXISTS(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Ebool_2ERES__EXISTS__UNIQUE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Ebool_2ERES__FORALL,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2ERES__FORALL(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Ebool_2ERES__SELECT,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2ERES__SELECT(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),A_27a))) ) ).
fof(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
fof(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
fof(mem_c_2Ebool_2ETYPE__DEFINITION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,A_27a),bool))) ) ) ).
fof(mem_c_2Ebool_2E_5C_2F,axiom,
mem(c_2Ebool_2E_5C_2F,arr(bool,arr(bool,bool))) ).
fof(ax_or_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_5C_2F,Q),R))
<=> ( p(Q)
| p(R) ) ) ) ) ).
fof(mem_c_2Ebool_2Eitself__case,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ebool_2Eitself__case(A_27a,A_27b),arr(ty_2Ebool_2Eitself(A_27a),arr(A_27b,A_27b))) ) ) ).
fof(mem_c_2Ebool_2Eliteral__case,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ebool_2Eliteral__case(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(A_27a,A_27b))) ) ) ).
fof(mem_c_2Ebool_2Ethe__value,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ebool_2Ethe__value(A_27a),ty_2Ebool_2Eitself(A_27a)) ) ).
fof(mem_c_2Ebool_2E_7E,axiom,
mem(c_2Ebool_2E_7E,arr(bool,bool)) ).
fof(ax_neg_p,axiom,
! [Q] :
( mem(Q,bool)
=> ( p(ap(c_2Ebool_2E_7E,Q))
<=> ~ p(Q) ) ) ).
fof(ax_thm_2Ebool_2ET__DEF,axiom,
( $true
<=> i(bool) = i(bool) ) ).
fof(lameq_f1,axiom,
! [A_27a,V0P] : ap(f1(A_27a),V0P) = ap(ap(c_2Emin_2E_3D(arr(A_27a,bool)),V0P),k(A_27a,c_2Ebool_2ET)) ).
fof(ax_thm_2Ebool_2EFORALL__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2E_21(A_27a) = f1(A_27a) ) ).
fof(lameq_f2,axiom,
! [A_27a,V0P] : ap(f2(A_27a),V0P) = ap(V0P,ap(c_2Emin_2E_40(A_27a),V0P)) ).
fof(ax_thm_2Ebool_2EEXISTS__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2E_3F(A_27a) = f2(A_27a) ) ).
fof(lameq_f3,axiom,
! [V1t2] :
( mem(V1t2,bool)
=> ! [V0t1] :
( mem(V0t1,bool)
=> ! [V2t] : ap(f3(V1t2,V0t1),V2t) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,V0t1),ap(ap(c_2Emin_2E_3D_3D_3E,V1t2),V2t))),V2t) ) ) ).
fof(lameq_f4,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] : ap(f4(V0t1),V1t2) = ap(c_2Ebool_2E_21(bool),f3(V1t2,V0t1)) ) ).
fof(lameq_f5,axiom,
! [V0t1] : ap(f5,V0t1) = f4(V0t1) ).
fof(ax_thm_2Ebool_2EAND__DEF,axiom,
c_2Ebool_2E_2F_5C = f5 ).
fof(lameq_f6,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ! [V2t] : ap(f6(V0t1,V1t2),V2t) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,V0t1),V2t)),ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D_3D_3E,V1t2),V2t)),V2t)) ) ) ).
fof(lameq_f7,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] : ap(f7(V0t1),V1t2) = ap(c_2Ebool_2E_21(bool),f6(V0t1,V1t2)) ) ).
fof(lameq_f8,axiom,
! [V0t1] : ap(f8,V0t1) = f7(V0t1) ).
fof(ax_thm_2Ebool_2EOR__DEF,axiom,
c_2Ebool_2E_5C_2F = f8 ).
fof(ax_thm_2Ebool_2EF__DEF,axiom,
( $false
<=> ! [V0t] :
( mem(V0t,bool)
=> p(V0t) ) ) ).
fof(lameq_f9,axiom,
! [V0t] : ap(f9,V0t) = ap(ap(c_2Emin_2E_3D_3D_3E,V0t),c_2Ebool_2EF) ).
fof(ax_thm_2Ebool_2ENOT__DEF,axiom,
c_2Ebool_2E_7E = f9 ).
fof(lameq_f10,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] : ap(f10(A_27a,V0P,V1x),V2y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V0P,V1x)),ap(V0P,V2y))),ap(ap(c_2Emin_2E_3D(A_27a),V1x),V2y)) ) ) ).
fof(lameq_f11,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] : ap(f11(A_27a,V0P),V1x) = ap(c_2Ebool_2E_21(A_27a),f10(A_27a,V0P,V1x)) ) ).
fof(lameq_f12,axiom,
! [A_27a,V0P] : ap(f12(A_27a),V0P) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_3F(A_27a),V0P)),ap(c_2Ebool_2E_21(A_27a),f11(A_27a,V0P))) ).
fof(ax_thm_2Ebool_2EEXISTS__UNIQUE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2E_3F_21(A_27a) = f12(A_27a) ) ).
fof(lameq_f13,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x] : ap(f13(A_27b,A_27a,V0f),V1x) = ap(V0f,V1x) ) ).
fof(lameq_f14,axiom,
! [A_27b,A_27a,V0f] : ap(f14(A_27b,A_27a),V0f) = f13(A_27b,A_27a,V0f) ).
fof(ax_thm_2Ebool_2ELET__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Ebool_2ELET(A_27a,A_27b) = f14(A_27b,A_27a) ) ) ).
fof(lameq_f15,axiom,
! [A_27a,V1t1] :
( mem(V1t1,A_27a)
=> ! [V2t2] :
( mem(V2t2,A_27a)
=> ! [V0t] :
( mem(V0t,bool)
=> ! [V3x] : ap(f15(A_27a,V1t1,V2t2,V0t),V3x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(bool),V0t),c_2Ebool_2ET)),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V1t1))),ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(bool),V0t),c_2Ebool_2EF)),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V2t2))) ) ) ) ).
fof(lameq_f16,axiom,
! [A_27a,V0t] :
( mem(V0t,bool)
=> ! [V1t1] :
( mem(V1t1,A_27a)
=> ! [V2t2] : ap(f16(A_27a,V0t,V1t1),V2t2) = ap(c_2Emin_2E_40(A_27a),f15(A_27a,V1t1,V2t2,V0t)) ) ) ).
fof(lameq_f17,axiom,
! [A_27a,V0t] :
( mem(V0t,bool)
=> ! [V1t1] : ap(f17(A_27a,V0t),V1t1) = f16(A_27a,V0t,V1t1) ) ).
fof(lameq_f18,axiom,
! [A_27a,V0t] : ap(f18(A_27a),V0t) = f17(A_27a,V0t) ).
fof(ax_thm_2Ebool_2ECOND__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2ECOND(A_27a) = f18(A_27a) ) ).
fof(lameq_f19,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x1] :
( mem(V1x1,A_27a)
=> ! [V2x2] : ap(f19(A_27b,A_27a,V0f,V1x1),V2x2) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(A_27b),ap(V0f,V1x1)),ap(V0f,V2x2))),ap(ap(c_2Emin_2E_3D(A_27a),V1x1),V2x2)) ) ) ).
fof(lameq_f20,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x1] : ap(f20(A_27b,A_27a,V0f),V1x1) = ap(c_2Ebool_2E_21(A_27a),f19(A_27b,A_27a,V0f,V1x1)) ) ).
fof(lameq_f21,axiom,
! [A_27b,A_27a,V0f] : ap(f21(A_27b,A_27a),V0f) = ap(c_2Ebool_2E_21(A_27a),f20(A_27b,A_27a,V0f)) ).
fof(ax_thm_2Ebool_2EONE__ONE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Ebool_2EONE__ONE(A_27a,A_27b) = f21(A_27b,A_27a) ) ) ).
fof(lameq_f22,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1y] :
( mem(V1y,A_27b)
=> ! [V2x] : ap(f22(A_27b,A_27a,V0f,V1y),V2x) = ap(ap(c_2Emin_2E_3D(A_27b),V1y),ap(V0f,V2x)) ) ) ).
fof(lameq_f23,axiom,
! [A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1y] : ap(f23(A_27a,A_27b,V0f),V1y) = ap(c_2Ebool_2E_3F(A_27a),f22(A_27b,A_27a,V0f,V1y)) ) ).
fof(lameq_f24,axiom,
! [A_27a,A_27b,V0f] : ap(f24(A_27a,A_27b),V0f) = ap(c_2Ebool_2E_21(A_27b),f23(A_27a,A_27b,V0f)) ).
fof(ax_thm_2Ebool_2EONTO__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Ebool_2EONTO(A_27a,A_27b) = f24(A_27a,A_27b) ) ) ).
fof(lameq_f25,axiom,
! [A_27a,A_27b,V1rep] :
( mem(V1rep,arr(A_27b,A_27a))
=> ! [V2x_27] :
( mem(V2x_27,A_27b)
=> ! [V3x_27_27] : ap(f25(A_27a,A_27b,V1rep,V2x_27),V3x_27_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Emin_2E_3D(A_27a),ap(V1rep,V2x_27)),ap(V1rep,V3x_27_27))),ap(ap(c_2Emin_2E_3D(A_27b),V2x_27),V3x_27_27)) ) ) ).
fof(lameq_f26,axiom,
! [A_27a,A_27b,V1rep] :
( mem(V1rep,arr(A_27b,A_27a))
=> ! [V2x_27] : ap(f26(A_27a,A_27b,V1rep),V2x_27) = ap(c_2Ebool_2E_21(A_27b),f25(A_27a,A_27b,V1rep,V2x_27)) ) ).
fof(lameq_f27,axiom,
! [A_27a,A_27b,V1rep] :
( mem(V1rep,arr(A_27b,A_27a))
=> ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5x_27] : ap(f27(A_27a,A_27b,V1rep,V4x),V5x_27) = ap(ap(c_2Emin_2E_3D(A_27a),V4x),ap(V1rep,V5x_27)) ) ) ).
fof(lameq_f28,axiom,
! [A_27b,A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1rep] :
( mem(V1rep,arr(A_27b,A_27a))
=> ! [V4x] : ap(f28(A_27b,A_27a,V0P,V1rep),V4x) = ap(ap(c_2Emin_2E_3D(bool),ap(V0P,V4x)),ap(c_2Ebool_2E_3F(A_27b),f27(A_27a,A_27b,V1rep,V4x))) ) ) ).
fof(lameq_f29,axiom,
! [A_27b,A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1rep] : ap(f29(A_27b,A_27a,V0P),V1rep) = ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Ebool_2E_21(A_27b),f26(A_27a,A_27b,V1rep))),ap(c_2Ebool_2E_21(A_27a),f28(A_27b,A_27a,V0P,V1rep))) ) ).
fof(lameq_f30,axiom,
! [A_27b,A_27a,V0P] : ap(f30(A_27b,A_27a),V0P) = f29(A_27b,A_27a,V0P) ).
fof(ax_thm_2Ebool_2ETYPE__DEFINITION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b) = f30(A_27b,A_27a) ) ) ).
fof(ax_thm_2Ebool_2EBOOL__CASES__AX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
<=> $true )
| ( p(V0t)
<=> $false ) ) ) ).
fof(lameq_f31,axiom,
! [A_27b,A_27a,V0t] :
( mem(V0t,arr(A_27a,A_27b))
=> ! [V1x] : ap(f31(A_27b,A_27a,V0t),V1x) = ap(V0t,V1x) ) ).
fof(ax_thm_2Ebool_2EETA__AX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t] :
( mem(V0t,arr(A_27a,A_27b))
=> f31(A_27b,A_27a,V0t) = V0t ) ) ) ).
fof(ax_thm_2Ebool_2ESELECT__AX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( p(ap(V0P,V1x))
=> p(ap(V0P,ap(c_2Emin_2E_40(A_27a),V0P))) ) ) ) ) ).
fof(ax_thm_2Ebool_2EINFINITY__AX,axiom,
? [V0f] :
( mem(V0f,arr(ind,ind))
& p(ap(c_2Ebool_2EONE__ONE(ind,ind),V0f))
& ~ p(ap(c_2Ebool_2EONTO(ind,ind),V0f)) ) ).
fof(ax_thm_2Ebool_2Eliteral__case__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Ebool_2Eliteral__case(A_27a,A_27b) = f14(A_27b,A_27a) ) ) ).
fof(lameq_f32,axiom,
! [A_27a,V0x] :
( mem(V0x,A_27a)
=> ! [V1f] : ap(f32(A_27a,V0x),V1f) = ap(V1f,V0x) ) ).
fof(lameq_f33,axiom,
! [A_27a,V0x] : ap(f33(A_27a),V0x) = f32(A_27a,V0x) ).
fof(ax_thm_2Ebool_2EIN__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2EIN(A_27a) = f33(A_27a) ) ).
fof(lameq_f34,axiom,
! [A_27a,V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m] :
( mem(V1m,arr(A_27a,bool))
=> ! [V2x] : ap(f34(A_27a,V0p,V1m),V2x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p)),ap(V1m,V2x)) ) ) ).
fof(lameq_f35,axiom,
! [A_27a,V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m] : ap(f35(A_27a,V0p),V1m) = ap(c_2Ebool_2E_21(A_27a),f34(A_27a,V0p,V1m)) ) ).
fof(lameq_f36,axiom,
! [A_27a,V0p] : ap(f36(A_27a),V0p) = f35(A_27a,V0p) ).
fof(ax_thm_2Ebool_2ERES__FORALL__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2ERES__FORALL(A_27a) = f36(A_27a) ) ).
fof(lameq_f37,axiom,
! [A_27a,V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m] :
( mem(V1m,arr(A_27a,bool))
=> ! [V2x] : ap(f37(A_27a,V0p,V1m),V2x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p)),ap(V1m,V2x)) ) ) ).
fof(lameq_f38,axiom,
! [A_27a,V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m] : ap(f38(A_27a,V0p),V1m) = ap(c_2Ebool_2E_3F(A_27a),f37(A_27a,V0p,V1m)) ) ).
fof(lameq_f39,axiom,
! [A_27a,V0p] : ap(f39(A_27a),V0p) = f38(A_27a,V0p) ).
fof(ax_thm_2Ebool_2ERES__EXISTS__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2ERES__EXISTS(A_27a) = f39(A_27a) ) ).
fof(lameq_f40,axiom,
! [A_27a,V1m] :
( mem(V1m,arr(A_27a,bool))
=> ! [V2x] : ap(f40(A_27a,V1m),V2x) = ap(V1m,V2x) ) ).
fof(lameq_f41,axiom,
! [A_27a,V1m] :
( mem(V1m,arr(A_27a,bool))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] : ap(f41(A_27a,V1m,V3x),V4y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V1m,V3x)),ap(V1m,V4y))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V4y)) ) ) ).
fof(lameq_f42,axiom,
! [A_27a,V1m] :
( mem(V1m,arr(A_27a,bool))
=> ! [V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V3x] : ap(f42(A_27a,V1m,V0p),V3x) = ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0p),f41(A_27a,V1m,V3x)) ) ) ).
fof(lameq_f43,axiom,
! [A_27a,V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m] : ap(f43(A_27a,V0p),V1m) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0p),f40(A_27a,V1m))),ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0p),f42(A_27a,V1m,V0p))) ) ).
fof(lameq_f44,axiom,
! [A_27a,V0p] : ap(f44(A_27a),V0p) = f43(A_27a,V0p) ).
fof(ax_thm_2Ebool_2ERES__EXISTS__UNIQUE__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a) = f44(A_27a) ) ).
fof(lameq_f45,axiom,
! [A_27a,V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m] : ap(f45(A_27a,V0p),V1m) = ap(c_2Emin_2E_40(A_27a),f37(A_27a,V0p,V1m)) ) ).
fof(lameq_f46,axiom,
! [A_27a,V0p] : ap(f46(A_27a),V0p) = f45(A_27a,V0p) ).
fof(ax_thm_2Ebool_2ERES__SELECT__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2ERES__SELECT(A_27a) = f46(A_27a) ) ).
fof(ax_thm_2Ebool_2EBOUNDED__DEF,axiom,
c_2Ebool_2EBOUNDED = k(bool,c_2Ebool_2ET) ).
fof(ax_thm_2Ebool_2EDATATYPE__TAG__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ebool_2EDATATYPE(A_27a) = k(A_27a,c_2Ebool_2ET) ) ).
fof(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
fof(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
=> p(V1t2) )
=> ( ( p(V1t2)
=> p(V0t1) )
=> ( p(V0t1)
<=> p(V1t2) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( $false
=> p(V0t) ) ) ).
fof(lameq_f47,axiom,
! [A_27b,A_27a,V0M] :
( mem(V0M,arr(A_27a,A_27b))
=> ! [V1x] : ap(f47(A_27b,A_27a,V0M),V1x) = ap(V0M,V1x) ) ).
fof(conj_thm_2Ebool_2EETA__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0M] :
( mem(V0M,arr(A_27a,A_27b))
=> f47(A_27b,A_27a,V0M) = V0M ) ) ) ).
fof(conj_thm_2Ebool_2EEXCLUDED__MIDDLE,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( p(V0t)
| ~ p(V0t) ) ) ).
fof(lameq_f48,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V2x] : ap(f48(A_27b,A_27a,V0f),V2x) = ap(V0f,V2x) ) ).
fof(conj_thm_2Ebool_2EBETA__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1y] :
( mem(V1y,A_27a)
=> ap(f48(A_27b,A_27a,V0f),V1y) = ap(V0f,V1y) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELET__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V0f),V1x) = ap(V0f,V1x) ) ) ) ) ).
fof(conj_thm_2Ebool_2EFORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,bool))
=> ( p(ap(c_2Ebool_2E_21(A_27a),V0f))
<=> ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(V0f,V1x)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EEXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,bool))
=> ( p(ap(c_2Ebool_2E_3F(A_27a),V0f))
<=> ? [V1x] :
( mem(V1x,A_27a)
& p(ap(V0f,V1x)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EABS__SIMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0t1] :
( mem(V0t1,A_27a)
=> ! [V1t2] :
( mem(V1t2,A_27b)
=> ap(k(A_27b,V0t1),V1t2) = V0t1 ) ) ) ) ).
fof(conj_thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,bool)
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> p(V0t) )
<=> p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2EEXISTS__SIMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,bool)
=> ( ? [V1x] :
( mem(V1x,A_27a)
& p(V0t) )
<=> p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2EAND__INTRO__THM,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( p(V0t1)
=> ( p(V1t2)
=> ( p(V0t1)
& p(V1t2) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EAND1__THM,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
& p(V1t2) )
=> p(V0t1) ) ) ) ).
fof(conj_thm_2Ebool_2EAND2__THM,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
& p(V1t2) )
=> p(V1t2) ) ) ) ).
fof(conj_thm_2Ebool_2ECONJ__SYM,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
& p(V1t2) )
<=> ( p(V1t2)
& p(V0t1) ) ) ) ) ).
fof(conj_thm_2Ebool_2ECONJ__COMM,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
& p(V1t2) )
<=> ( p(V1t2)
& p(V0t1) ) ) ) ) ).
fof(conj_thm_2Ebool_2ECONJ__ASSOC,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ! [V2t3] :
( mem(V2t3,bool)
=> ( ( p(V0t1)
& p(V1t2)
& p(V2t3) )
<=> ( p(V0t1)
& p(V1t2)
& p(V2t3) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EOR__INTRO__THM1,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( p(V0t1)
=> ( p(V0t1)
| p(V1t2) ) ) ) ) ).
fof(conj_thm_2Ebool_2EOR__INTRO__THM2,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( p(V1t2)
=> ( p(V0t1)
| p(V1t2) ) ) ) ) ).
fof(conj_thm_2Ebool_2EOR__ELIM__THM,axiom,
! [V0t] :
( mem(V0t,bool)
=> ! [V1t1] :
( mem(V1t1,bool)
=> ! [V2t2] :
( mem(V2t2,bool)
=> ( ( p(V1t1)
| p(V2t2) )
=> ( ( p(V1t1)
=> p(V0t) )
=> ( ( p(V2t2)
=> p(V0t) )
=> p(V0t) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__F,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
=> $false )
=> ~ p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ p(V0t)
=> ( p(V0t)
=> $false ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__F,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ p(V0t)
=> ( p(V0t)
<=> $false ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__AND,axiom,
! [V0t] :
( mem(V0t,bool)
=> ~ ( p(V0t)
& ~ p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
& p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
& $true )
<=> p(V0t) )
& ( ( $false
& p(V0t) )
<=> $false )
& ( ( p(V0t)
& $false )
<=> $false )
& ( ( p(V0t)
& p(V0t) )
<=> p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
| p(V0t) )
<=> $true )
& ( ( p(V0t)
| $true )
<=> $true )
& ( ( $false
| p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
| $false )
<=> p(V0t) )
& ( ( p(V0t)
| p(V0t) )
<=> p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
=> p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
=> $true )
<=> $true )
& ( ( $false
=> p(V0t) )
<=> $true )
& ( ( p(V0t)
=> p(V0t) )
<=> $true )
& ( ( p(V0t)
=> $false )
<=> ~ p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t] :
( mem(V0t,bool)
=> ( ~ ~ p(V0t)
<=> p(V0t) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
fof(conj_thm_2Ebool_2EEQ__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> V0x = V0x ) ) ).
fof(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ( V0x = V0x
<=> $true ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( V0x = V1y
=> V1y = V0x ) ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( V0x = V1y
<=> V1y = V0x ) ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__EXT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27b))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ap(V0f,V2x) = ap(V1g,V2x) )
=> V0f = V1g ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EFUN__EQ__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27b))
=> ( V0f = V1g
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ap(V0f,V2x) = ap(V1g,V2x) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__TRANS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ! [V2z] :
( mem(V2z,A_27a)
=> ( ( V0x = V1y
& V1y = V2z )
=> V0x = V2z ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EBOOL__EQ__DISTINCT,axiom,
( ~ ( $true
<=> $false )
& ~ ( $false
<=> $true ) ) ).
fof(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
<=> p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
<=> $true )
<=> p(V0t) )
& ( ( $false
<=> p(V0t) )
<=> ~ p(V0t) )
& ( ( p(V0t)
<=> $false )
<=> ~ p(V0t) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t1] :
( mem(V0t1,A_27a)
=> ! [V1t2] :
( mem(V1t2,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),c_2Ebool_2ET),V0t1),V1t2) = V0t1
& ap(ap(ap(c_2Ebool_2ECOND(A_27a),c_2Ebool_2EF),V0t1),V1t2) = V1t2 ) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0b] :
( mem(V0b,bool)
=> ! [V1t] :
( mem(V1t,A_27a)
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),V0b),V1t),V1t) = V1t ) ) ) ).
fof(lameq_f49,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] : ap(f49(A_27a,V0P),V1x) = ap(V0P,V1x) ) ).
fof(conj_thm_2Ebool_2ESELECT__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(V0P,ap(c_2Emin_2E_40(A_27a),f49(A_27a,V0P))))
<=> ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) ) ) ) ) ).
fof(lameq_f50,axiom,
! [A_27a,V0x] :
( mem(V0x,A_27a)
=> ! [V1y] : ap(f50(A_27a,V0x),V1y) = ap(ap(c_2Emin_2E_3D(A_27a),V1y),V0x) ) ).
fof(conj_thm_2Ebool_2ESELECT__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Emin_2E_40(A_27a),f50(A_27a,V0x)) = V0x ) ) ).
fof(lameq_f51,axiom,
! [A_27a,V0x] :
( mem(V0x,A_27a)
=> ! [V1y] : ap(f51(A_27a,V0x),V1y) = ap(ap(c_2Emin_2E_3D(A_27a),V0x),V1y) ) ).
fof(conj_thm_2Ebool_2ESELECT__REFL__2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Emin_2E_40(A_27a),f51(A_27a,V0x)) = V0x ) ) ).
fof(conj_thm_2Ebool_2ESELECT__UNIQUE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( ! [V2y] :
( mem(V2y,A_27a)
=> ( p(ap(V0P,V2y))
<=> V2y = V1x ) )
=> ap(c_2Emin_2E_40(A_27a),V0P) = V1x ) ) ) ) ).
fof(conj_thm_2Ebool_2ESELECT__ELIM__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(V0P,V3x))
=> p(ap(V1Q,V3x)) ) ) )
=> p(ap(V1Q,ap(c_2Emin_2E_40(A_27a),V0P))) ) ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ~ ! [V1x] :
( mem(V1x,A_27a)
=> p(ap(V0P,V1x)) )
<=> ? [V2x] :
( mem(V2x,A_27a)
& ~ p(ap(V0P,V2x)) ) ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ~ ? [V1x] :
( mem(V1x,A_27a)
& p(ap(V0P,V1x)) )
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ~ p(ap(V0P,V2x)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EFORALL__AND__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0P,V2x))
& p(ap(V1Q,V2x)) ) )
<=> ( ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V0P,V3x)) )
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(V1Q,V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V0P,V2x)) )
& p(V1Q) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(V0P,V3x))
& p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__AND__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( p(V0P)
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V1Q,V2x)) ) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(V0P)
& p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EEXISTS__OR__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ? [V2x] :
( mem(V2x,A_27a)
& ( p(ap(V0P,V2x))
| p(ap(V1Q,V2x)) ) )
<=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V0P,V3x)) )
| ? [V4x] :
( mem(V4x,A_27a)
& p(ap(V1Q,V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
| p(V1Q) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& ( p(ap(V0P,V3x))
| p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( p(V0P)
| ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V1Q,V2x)) ) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& ( p(V0P)
| p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EBOTH__EXISTS__AND__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(V0P)
& p(V1Q) )
<=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(V0P) )
& ? [V4x] :
( mem(V4x,A_27a)
& p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x))
& p(V1Q) )
<=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V0P,V3x)) )
& p(V1Q) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(V0P)
& p(ap(V1Q,V2x)) )
<=> ( p(V0P)
& ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EBOTH__FORALL__OR__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(V0P)
| p(V1Q) ) )
<=> ( ! [V3x] :
( mem(V3x,A_27a)
=> p(V0P) )
| ! [V4x] :
( mem(V4x,A_27a)
=> p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__FORALL__OR__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0Q] :
( mem(V0Q,bool)
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V1P,V2x))
| p(V0Q) ) )
<=> ( ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V1P,V3x)) )
| p(V0Q) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(V0P)
| p(ap(V1Q,V2x)) ) )
<=> ( p(V0P)
| ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EBOTH__FORALL__IMP__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(V0P)
=> p(V1Q) ) )
<=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(V0P) )
=> ! [V4x] :
( mem(V4x,A_27a)
=> p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__FORALL__IMP__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0P,V2x))
=> p(V1Q) ) )
<=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V0P,V3x)) )
=> p(V1Q) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__FORALL__IMP__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(V0P)
=> p(ap(V1Q,V2x)) ) )
<=> ( p(V0P)
=> ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EBOTH__EXISTS__IMP__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& ( p(V0P)
=> p(V1Q) ) )
<=> ( ! [V3x] :
( mem(V3x,A_27a)
=> p(V0P) )
=> ? [V4x] :
( mem(V4x,A_27a)
& p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__EXISTS__IMP__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& ( p(ap(V0P,V2x))
=> p(V1Q) ) )
<=> ( ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V0P,V3x)) )
=> p(V1Q) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__EXISTS__IMP__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ? [V2x] :
( mem(V2x,A_27a)
& ( p(V0P)
=> p(ap(V1Q,V2x)) ) )
<=> ( p(V0P)
=> ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EOR__IMP__THM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
<=> ( p(V1B)
| p(V0A) ) )
<=> ( p(V1B)
=> p(V0A) ) ) ) ) ).
fof(conj_thm_2Ebool_2ENOT__IMP,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ~ ( p(V0A)
=> p(V1B) )
<=> ( p(V0A)
& ~ p(V1B) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__ASSOC,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ! [V2C] :
( mem(V2C,bool)
=> ( ( p(V0A)
| p(V1B)
| p(V2C) )
<=> ( p(V0A)
| p(V1B)
| p(V2C) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
| p(V1B) )
<=> ( p(V1B)
| p(V0A) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__COMM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
| p(V1B) )
<=> ( p(V1B)
| p(V0A) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDE__MORGAN__THM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( p(V0A)
& p(V1B) )
<=> ( ~ p(V0A)
| ~ p(V1B) ) )
& ( ~ ( p(V0A)
| p(V1B) )
<=> ( ~ p(V0A)
& ~ p(V1B) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__AND__OVER__OR,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ! [V2C] :
( mem(V2C,bool)
=> ( ( p(V0A)
& ( p(V1B)
| p(V2C) ) )
<=> ( ( p(V0A)
& p(V1B) )
| ( p(V0A)
& p(V2C) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__AND__OVER__OR,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ! [V2C] :
( mem(V2C,bool)
=> ( ( ( p(V1B)
| p(V2C) )
& p(V0A) )
<=> ( ( p(V1B)
& p(V0A) )
| ( p(V2C)
& p(V0A) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__OR__OVER__AND,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ! [V2C] :
( mem(V2C,bool)
=> ( ( p(V0A)
| ( p(V1B)
& p(V2C) ) )
<=> ( ( p(V0A)
| p(V1B) )
& ( p(V0A)
| p(V2C) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__OR__OVER__AND,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ! [V2C] :
( mem(V2C,bool)
=> ( ( ( p(V1B)
& p(V2C) )
| p(V0A) )
<=> ( ( p(V1B)
| p(V0A) )
& ( p(V2C)
| p(V0A) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__DISJ__THM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
=> p(V1B) )
<=> ( ~ p(V0A)
| p(V1B) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__IMP__THM,axiom,
! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ! [V2R] :
( mem(V2R,bool)
=> ( ( ( p(V0P)
| p(V1Q) )
=> p(V2R) )
<=> ( ( p(V0P)
=> p(V2R) )
& ( p(V1Q)
=> p(V2R) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__CONJ__THM,axiom,
! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ! [V2R] :
( mem(V2R,bool)
=> ( ( p(V0P)
=> ( p(V1Q)
& p(V2R) ) )
<=> ( ( p(V0P)
=> p(V1Q) )
& ( p(V0P)
=> p(V2R) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__F__EQ__F,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
=> $false )
<=> ( p(V0t)
<=> $false ) ) ) ).
fof(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ! [V2t3] :
( mem(V2t3,bool)
=> ( ( p(V0t1)
=> ( p(V1t2)
=> p(V2t3) ) )
<=> ( ( p(V0t1)
& p(V1t2) )
=> p(V2t3) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__IMP__THM,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
<=> p(V1t2) )
<=> ( ( p(V0t1)
=> p(V1t2) )
& ( p(V1t2)
=> p(V0t1) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EEQ__EXPAND,axiom,
! [V0t1] :
( mem(V0t1,bool)
=> ! [V1t2] :
( mem(V1t2,bool)
=> ( ( p(V0t1)
<=> p(V1t2) )
<=> ( ( p(V0t1)
& p(V1t2) )
| ( ~ p(V0t1)
& ~ p(V1t2) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__RATOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0b] :
( mem(V0b,bool)
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ap(ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,A_27b)),V0b),V1f),V2g),V3x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),V0b),ap(V1f,V3x)),ap(V2g,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__RAND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1b] :
( mem(V1b,bool)
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ap(V0f,ap(ap(ap(c_2Ebool_2ECOND(A_27a),V1b),V2x),V3y)) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),V1b),ap(V0f,V2x)),ap(V0f,V3y)) ) ) ) ) ) ) ).
fof(lameq_f52,axiom,
! [A_27b,A_27a,V0b] :
( mem(V0b,bool)
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g] :
( mem(V2g,arr(A_27a,A_27b))
=> ! [V3x] : ap(f52(A_27b,A_27a,V0b,V1f,V2g),V3x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),V0b),ap(V1f,V3x)),ap(V2g,V3x)) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__ABS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0b] :
( mem(V0b,bool)
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2g] :
( mem(V2g,arr(A_27a,A_27b))
=> f52(A_27b,A_27a,V0b,V1f,V2g) = ap(ap(ap(c_2Ebool_2ECOND(arr(A_27a,A_27b)),V0b),V1f),V2g) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__EXPAND,axiom,
! [V0b] :
( mem(V0b,bool)
=> ! [V1t1] :
( mem(V1t1,bool)
=> ! [V2t2] :
( mem(V2t2,bool)
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0b),V1t1),V2t2))
<=> ( ( ~ p(V0b)
| p(V1t1) )
& ( p(V0b)
| p(V2t2) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__EXPAND__IMP,axiom,
! [V0b] :
( mem(V0b,bool)
=> ! [V1t1] :
( mem(V1t1,bool)
=> ! [V2t2] :
( mem(V2t2,bool)
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0b),V1t1),V2t2))
<=> ( ( p(V0b)
=> p(V1t1) )
& ( ~ p(V0b)
=> p(V2t2) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__EXPAND__OR,axiom,
! [V0b] :
( mem(V0b,bool)
=> ! [V1t1] :
( mem(V1t1,bool)
=> ! [V2t2] :
( mem(V2t2,bool)
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0b),V1t1),V2t2))
<=> ( ( p(V0b)
& p(V1t1) )
| ( ~ p(V0b)
& p(V2t2) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ETYPE__DEFINITION__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1rep] :
( mem(V1rep,arr(A_27b,A_27a))
=> ( p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),V0P),V1rep))
<=> ( ! [V2x_27] :
( mem(V2x_27,A_27b)
=> ! [V3x_27_27] :
( mem(V3x_27_27,A_27b)
=> ( ap(V1rep,V2x_27) = ap(V1rep,V3x_27_27)
=> V2x_27 = V3x_27_27 ) ) )
& ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(V0P,V4x))
<=> ? [V5x_27] :
( mem(V5x_27,A_27b)
& V4x = ap(V1rep,V5x_27) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EONTO__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( p(ap(c_2Ebool_2EONTO(A_27a,A_27b),V0f))
<=> ! [V1y] :
( mem(V1y,A_27b)
=> ? [V2x] :
( mem(V2x,A_27a)
& V1y = ap(V0f,V2x) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EONE__ONE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( p(ap(c_2Ebool_2EONE__ONE(A_27a,A_27b),V0f))
<=> ! [V1x1] :
( mem(V1x1,A_27a)
=> ! [V2x2] :
( mem(V2x2,A_27a)
=> ( ap(V0f,V1x1) = ap(V0f,V2x2)
=> V1x1 = V2x2 ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EABS__REP__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ? [V1rep] :
( mem(V1rep,arr(A_27b,A_27a))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,A_27b),V0P),V1rep)) )
=> ? [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
& ? [V3abs] :
( mem(V3abs,arr(A_27a,A_27b))
& ! [V4a] :
( mem(V4a,A_27b)
=> ap(V3abs,ap(V2rep,V4a)) = V4a )
& ! [V5r] :
( mem(V5r,A_27a)
=> ( p(ap(V0P,V5r))
<=> ap(V2rep,ap(V3abs,V5r)) = V5r ) ) ) ) ) ) ) ) ).
fof(lameq_f53,axiom,
! [A_27b,A_27a,V1N] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V3x] : ap(f53(A_27b,A_27a,V1N),V3x) = ap(V1N,V3x) ) ).
fof(lameq_f54,axiom,
! [A_27b,A_27a,V1N] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V0P] :
( mem(V0P,arr(A_27b,bool))
=> ! [V4x] : ap(f54(A_27b,A_27a,V1N,V0P),V4x) = ap(V0P,ap(V1N,V4x)) ) ) ).
fof(conj_thm_2Ebool_2ELET__RAND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27b,bool))
=> ! [V1N] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V2M] :
( mem(V2M,A_27a)
=> ( p(ap(V0P,ap(ap(c_2Ebool_2ELET(A_27a,A_27b),f53(A_27b,A_27a,V1N)),V2M)))
<=> p(ap(ap(c_2Ebool_2ELET(A_27a,bool),f54(A_27b,A_27a,V1N,V0P)),V2M)) ) ) ) ) ) ) ).
fof(lameq_f55,axiom,
! [A_27c,A_27b,A_27a,V0N] :
( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V3x] : ap(f55(A_27c,A_27b,A_27a,V0N),V3x) = ap(V0N,V3x) ) ).
fof(lameq_f56,axiom,
! [A_27b,A_27c,A_27a,V0N] :
( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V2b] :
( mem(V2b,A_27b)
=> ! [V4x] : ap(f56(A_27b,A_27c,A_27a,V0N,V2b),V4x) = ap(ap(V0N,V4x),V2b) ) ) ).
fof(conj_thm_2Ebool_2ELET__RATOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0N] :
( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1M] :
( mem(V1M,A_27a)
=> ! [V2b] :
( mem(V2b,A_27b)
=> ap(ap(ap(c_2Ebool_2ELET(A_27a,arr(A_27b,A_27c)),f55(A_27c,A_27b,A_27a,V0N)),V1M),V2b) = ap(ap(c_2Ebool_2ELET(A_27a,A_27c),f56(A_27b,A_27c,A_27a,V0N,V2b)),V1M) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ESWAP__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27b)
=> p(ap(ap(V0P,V1x),V2y)) ) )
<=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(V0P,V4x),V3y)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ESWAP__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( ? [V1x] :
( mem(V1x,A_27a)
& ? [V2y] :
( mem(V2y,A_27b)
& p(ap(ap(V0P,V1x),V2y)) ) )
<=> ? [V3y] :
( mem(V3y,A_27b)
& ? [V4x] :
( mem(V4x,A_27a)
& p(ap(ap(V0P,V4x),V3y)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EEXISTS__UNIQUE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(c_2Ebool_2E_3F_21(A_27a),f49(A_27a,V0P)))
<=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
& ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ( p(ap(V0P,V3x))
& p(ap(V0P,V4y)) )
=> V3x = V4y ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELET__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2M] :
( mem(V2M,A_27a)
=> ! [V3N] :
( mem(V3N,A_27a)
=> ( ( V2M = V3N
& ! [V4x] :
( mem(V4x,A_27a)
=> ( V4x = V3N
=> ap(V0f,V4x) = ap(V1g,V4x) ) ) )
=> ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V0f),V2M) = ap(ap(c_2Ebool_2ELET(A_27a,A_27b),V1g),V3N) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( p(V0x)
<=> p(V1x_27) )
& ( p(V1x_27)
=> ( p(V2y)
<=> p(V3y_27) ) ) )
=> ( ( p(V0x)
=> p(V2y) )
<=> ( p(V1x_27)
=> p(V3y_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EAND__CONG,axiom,
! [V0P] :
( mem(V0P,bool)
=> ! [V1P_27] :
( mem(V1P_27,bool)
=> ! [V2Q] :
( mem(V2Q,bool)
=> ! [V3Q_27] :
( mem(V3Q_27,bool)
=> ( ( ( p(V2Q)
=> ( p(V0P)
<=> p(V1P_27) ) )
& ( p(V1P_27)
=> ( p(V2Q)
<=> p(V3Q_27) ) ) )
=> ( ( p(V0P)
& p(V2Q) )
<=> ( p(V1P_27)
& p(V3Q_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__AND__CONG,axiom,
! [V0P] :
( mem(V0P,bool)
=> ! [V1P_27] :
( mem(V1P_27,bool)
=> ! [V2Q] :
( mem(V2Q,bool)
=> ! [V3Q_27] :
( mem(V3Q_27,bool)
=> ( ( ( p(V0P)
<=> p(V1P_27) )
& ( p(V1P_27)
=> ( p(V2Q)
<=> p(V3Q_27) ) ) )
=> ( ( p(V0P)
& p(V2Q) )
<=> ( p(V1P_27)
& p(V3Q_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EOR__CONG,axiom,
! [V0P] :
( mem(V0P,bool)
=> ! [V1P_27] :
( mem(V1P_27,bool)
=> ! [V2Q] :
( mem(V2Q,bool)
=> ! [V3Q_27] :
( mem(V3Q_27,bool)
=> ( ( ( ~ p(V2Q)
=> ( p(V0P)
<=> p(V1P_27) ) )
& ( ~ p(V1P_27)
=> ( p(V2Q)
<=> p(V3Q_27) ) ) )
=> ( ( p(V0P)
| p(V2Q) )
<=> ( p(V1P_27)
| p(V3Q_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__OR__CONG,axiom,
! [V0P] :
( mem(V0P,bool)
=> ! [V1P_27] :
( mem(V1P_27,bool)
=> ! [V2Q] :
( mem(V2Q,bool)
=> ! [V3Q_27] :
( mem(V3Q_27,bool)
=> ( ( ( p(V0P)
<=> p(V1P_27) )
& ( ~ p(V1P_27)
=> ( p(V2Q)
<=> p(V3Q_27) ) ) )
=> ( ( p(V0P)
| p(V2Q) )
<=> ( p(V1P_27)
| p(V3Q_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ECOND__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3x_27] :
( mem(V3x_27,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ! [V5y_27] :
( mem(V5y_27,A_27a)
=> ( ( ( p(V0P)
<=> p(V1Q) )
& ( p(V1Q)
=> V2x = V3x_27 )
& ( ~ p(V1Q)
=> V4y = V5y_27 ) )
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),V0P),V2x),V4y) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),V1Q),V3x_27),V5y_27) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERES__FORALL__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,bool))
=> ! [V3g] :
( mem(V3g,arr(A_27a,bool))
=> ( V0P = V1Q
=> ( ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1Q))
=> ( p(ap(V2f,V4x))
<=> p(ap(V3g,V4x)) ) ) )
=> ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),V2f))
<=> p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V1Q),V3g)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERES__EXISTS__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V2f] :
( mem(V2f,arr(A_27a,bool))
=> ! [V3g] :
( mem(V3g,arr(A_27a,bool))
=> ( V0P = V1Q
=> ( ! [V4x] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V1Q))
=> ( p(ap(V2f,V4x))
<=> p(ap(V3g,V4x)) ) ) )
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),V2f))
<=> p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V1Q),V3g)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EMONO__AND,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1y] :
( mem(V1y,bool)
=> ! [V2z] :
( mem(V2z,bool)
=> ! [V3w] :
( mem(V3w,bool)
=> ( ( ( p(V0x)
=> p(V1y) )
& ( p(V2z)
=> p(V3w) ) )
=> ( ( p(V0x)
& p(V2z) )
=> ( p(V1y)
& p(V3w) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EMONO__OR,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1y] :
( mem(V1y,bool)
=> ! [V2z] :
( mem(V2z,bool)
=> ! [V3w] :
( mem(V3w,bool)
=> ( ( ( p(V0x)
=> p(V1y) )
& ( p(V2z)
=> p(V3w) ) )
=> ( ( p(V0x)
| p(V2z) )
=> ( p(V1y)
| p(V3w) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EMONO__IMP,axiom,
! [V0y] :
( mem(V0y,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ! [V2z] :
( mem(V2z,bool)
=> ! [V3w] :
( mem(V3w,bool)
=> ( ( ( p(V0y)
=> p(V1x) )
& ( p(V2z)
=> p(V3w) ) )
=> ( ( p(V1x)
=> p(V2z) )
=> ( p(V0y)
=> p(V3w) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EMONO__NOT,axiom,
! [V0y] :
( mem(V0y,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ( ( p(V0y)
=> p(V1x) )
=> ( ~ p(V1x)
=> ~ p(V0y) ) ) ) ) ).
fof(conj_thm_2Ebool_2EMONO__NOT__EQ,axiom,
! [V0y] :
( mem(V0y,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ( ( p(V0y)
=> p(V1x) )
<=> ( ~ p(V1x)
=> ~ p(V0y) ) ) ) ) ).
fof(conj_thm_2Ebool_2EMONO__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0P,V2x))
=> p(ap(V1Q,V2x)) ) )
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V0P,V3x)) )
=> ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(V1Q,V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EMONO__EXISTS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0P,V2x))
=> p(ap(V1Q,V2x)) ) )
=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V0P,V3x)) )
=> ? [V4x] :
( mem(V4x,A_27a)
& p(ap(V1Q,V4x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EMONO__COND,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1y] :
( mem(V1y,bool)
=> ! [V2z] :
( mem(V2z,bool)
=> ! [V3w] :
( mem(V3w,bool)
=> ! [V4b] :
( mem(V4b,bool)
=> ( ( p(V0x)
=> p(V1y) )
=> ( ( p(V2z)
=> p(V3w) )
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V4b),V0x),V2z))
=> p(ap(ap(ap(c_2Ebool_2ECOND(bool),V4b),V1y),V3w)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EEXISTS__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ? [V1x] :
( mem(V1x,A_27a)
& V1x = V0a ) ) ) ).
fof(lameq_f57,axiom,
! [A_27a,V0a] :
( mem(V0a,A_27a)
=> ! [V1x] : ap(f57(A_27a,V0a),V1x) = ap(ap(c_2Emin_2E_3D(A_27a),V1x),V0a) ) ).
fof(conj_thm_2Ebool_2EEXISTS__UNIQUE__REFL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0a] :
( mem(V0a,A_27a)
=> p(ap(c_2Ebool_2E_3F_21(A_27a),f57(A_27a,V0a))) ) ) ).
fof(conj_thm_2Ebool_2EUNWIND__THM1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& V1a = V2x
& p(ap(V0P,V2x)) )
<=> p(ap(V0P,V1a)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EUNWIND__THM2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& V2x = V1a
& p(ap(V0P,V2x)) )
<=> p(ap(V0P,V1a)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EUNWIND__FORALL__THM1,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,bool))
=> ! [V1v] :
( mem(V1v,A_27a)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( V1v = V2x
=> p(ap(V0f,V2x)) ) )
<=> p(ap(V0f,V1v)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EUNWIND__FORALL__THM2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,bool))
=> ! [V1v] :
( mem(V1v,A_27a)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( V2x = V1v
=> p(ap(V0f,V2x)) ) )
<=> p(ap(V0f,V1v)) ) ) ) ) ).
fof(conj_thm_2Ebool_2ESKOLEM__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ? [V2y] :
( mem(V2y,A_27b)
& p(ap(ap(V0P,V1x),V2y)) ) )
<=> ? [V3f] :
( mem(V3f,arr(A_27a,A_27b))
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(V0P,V4x),ap(V3f,V4x))) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2Ebool__case__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0t1] :
( mem(V0t1,A_27a)
=> ! [V1t2] :
( mem(V1t2,A_27a)
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),c_2Ebool_2ET),V0t1),V1t2) = V0t1 ) )
& ! [V2t1] :
( mem(V2t1,A_27a)
=> ! [V3t2] :
( mem(V3t2,A_27a)
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),c_2Ebool_2EF),V2t1),V3t2) = V3t2 ) ) ) ) ).
fof(conj_thm_2Ebool_2Ebool__case__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0b] :
( mem(V0b,bool)
=> ! [V1t] :
( mem(V1t,A_27a)
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),V0b),V1t),V1t) = V1t ) ) ) ).
fof(conj_thm_2Ebool_2EboolAxiom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t1] :
( mem(V0t1,A_27a)
=> ! [V1t2] :
( mem(V1t2,A_27a)
=> ? [V2fn] :
( mem(V2fn,arr(bool,A_27a))
& ap(V2fn,c_2Ebool_2ET) = V0t1
& ap(V2fn,c_2Ebool_2EF) = V1t2 ) ) ) ) ).
fof(conj_thm_2Ebool_2Ebool__INDUCT,axiom,
! [V0P] :
( mem(V0P,arr(bool,bool))
=> ( ( p(ap(V0P,c_2Ebool_2ET))
& p(ap(V0P,c_2Ebool_2EF)) )
=> ! [V1b] :
( mem(V1b,bool)
=> p(ap(V0P,V1b)) ) ) ) ).
fof(conj_thm_2Ebool_2Ebool__case__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3x_27] :
( mem(V3x_27,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ! [V5y_27] :
( mem(V5y_27,A_27a)
=> ( ( ( p(V0P)
<=> p(V1Q) )
& ( p(V1Q)
=> V2x = V3x_27 )
& ( ~ p(V1Q)
=> V4y = V5y_27 ) )
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),V0P),V2x),V4y) = ap(ap(ap(c_2Ebool_2ECOND(A_27a),V1Q),V3x_27),V5y_27) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EFORALL__BOOL,axiom,
! [V0P] :
( mem(V0P,arr(bool,bool))
=> ( ! [V1b] :
( mem(V1b,bool)
=> p(ap(V0P,V1b)) )
<=> ( p(ap(V0P,c_2Ebool_2ET))
& p(ap(V0P,c_2Ebool_2EF)) ) ) ) ).
fof(lameq_f58,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V2x] : ap(f58(A_27a,V0P,V1Q),V2x) = ap(ap(c_2Ebool_2E_5C_2F,ap(V0P,V2x)),ap(V1Q,V2x)) ) ) ).
fof(lameq_f59,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V3x] : ap(f59(A_27a,V0P),V3x) = ap(V0P,V3x) ) ).
fof(lameq_f60,axiom,
! [A_27a,V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ! [V4x] : ap(f60(A_27a,V1Q),V4x) = ap(V1Q,V4x) ) ).
fof(conj_thm_2Ebool_2EUEXISTS__OR__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( p(ap(c_2Ebool_2E_3F_21(A_27a),f58(A_27a,V0P,V1Q)))
=> ( p(ap(c_2Ebool_2E_3F_21(A_27a),f59(A_27a,V0P)))
| p(ap(c_2Ebool_2E_3F_21(A_27a),f60(A_27a,V1Q))) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EUEXISTS__SIMP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t] :
( mem(V0t,bool)
=> ( p(ap(c_2Ebool_2E_3F_21(A_27a),k(A_27a,V0t)))
<=> ( p(V0t)
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27a)
=> V2x = V3y ) ) ) ) ) ) ).
fof(ax_thm_2Ebool_2ERES__ABSTRACT__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0p] :
( mem(V0p,arr(A_27a,bool))
=> ! [V1m] :
( mem(V1m,arr(A_27a,A_27b))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0p))
=> ap(ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V0p),V1m),V2x) = ap(V1m,V2x) ) ) ) )
& ! [V3p] :
( mem(V3p,arr(A_27a,bool))
=> ! [V4m1] :
( mem(V4m1,arr(A_27a,A_27b))
=> ! [V5m2] :
( mem(V5m2,arr(A_27a,A_27b))
=> ( ! [V6x] :
( mem(V6x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V6x),V3p))
=> ap(V4m1,V6x) = ap(V5m2,V6x) ) )
=> ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V3p),V4m1) = ap(ap(c_2Ebool_2ERES__ABSTRACT(A_27a,A_27b),V3p),V5m2) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERES__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),V1f))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
=> p(ap(V1f,V2x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERES__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),V1f))
<=> ? [V2x] :
( mem(V2x,A_27a)
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P))
& p(ap(V1f,V2x)) ) ) ) ) ) ).
fof(lameq_f61,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,bool))
=> ! [V2x] : ap(f61(A_27a,V1f),V2x) = ap(V1f,V2x) ) ).
fof(lameq_f62,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,bool))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] : ap(f62(A_27a,V1f,V3x),V4y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_2F_5C,ap(V1f,V3x)),ap(V1f,V4y))),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V4y)) ) ) ).
fof(lameq_f63,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(A_27a,bool))
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V3x] : ap(f63(A_27a,V1f,V0P),V3x) = ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),f62(A_27a,V1f,V3x)) ) ) ).
fof(conj_thm_2Ebool_2ERES__EXISTS__UNIQUE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS__UNIQUE(A_27a),V0P),V1f))
<=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),f61(A_27a,V1f)))
& p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),f63(A_27a,V1f,V0P))) ) ) ) ) ) ).
fof(lameq_f64,axiom,
! [A_27a,V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,bool))
=> ! [V2x] : ap(f64(A_27a,V0P,V1f),V2x) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0P)),ap(V1f,V2x)) ) ) ).
fof(conj_thm_2Ebool_2ERES__SELECT__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,bool))
=> ap(ap(c_2Ebool_2ERES__SELECT(A_27a),V0P),V1f) = ap(c_2Emin_2E_40(A_27a),f64(A_27a,V0P,V1f)) ) ) ) ).
fof(conj_thm_2Ebool_2ERES__FORALL__TRUE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__FORALL(A_27a),V0P),k(A_27a,c_2Ebool_2ET)))
<=> $true ) ) ) ).
fof(conj_thm_2Ebool_2ERES__EXISTS__FALSE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( p(ap(ap(c_2Ebool_2ERES__EXISTS(A_27a),V0P),k(A_27a,c_2Ebool_2EF)))
<=> $false ) ) ) ).
fof(lameq_f65,axiom,
! [V4b] : ap(f65,V4b) = ap(c_2Ebool_2E_7E,V4b) ).
fof(conj_thm_2Ebool_2EBOOL__FUN__CASES__THM,axiom,
! [V0f] :
( mem(V0f,arr(bool,bool))
=> ( V0f = k(bool,c_2Ebool_2ET)
| V0f = k(bool,c_2Ebool_2EF)
| V0f = i(bool)
| V0f = f65 ) ) ).
fof(conj_thm_2Ebool_2EBOOL__FUN__INDUCT,axiom,
! [V0P] :
( mem(V0P,arr(arr(bool,bool),bool))
=> ( ( p(ap(V0P,k(bool,c_2Ebool_2ET)))
& p(ap(V0P,k(bool,c_2Ebool_2EF)))
& p(ap(V0P,i(bool)))
& p(ap(V0P,f65)) )
=> ! [V5f] :
( mem(V5f,arr(bool,bool))
=> p(ap(V0P,V5f)) ) ) ) ).
fof(conj_thm_2Ebool_2Eliteral__case__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V0f),V1x) = ap(V0f,V1x) ) ) ) ) ).
fof(lameq_f66,axiom,
! [A_27b,A_27c,A_27a,V1N] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V0P] :
( mem(V0P,arr(A_27b,A_27c))
=> ! [V4x] : ap(f66(A_27b,A_27c,A_27a,V1N,V0P),V4x) = ap(V0P,ap(V1N,V4x)) ) ) ).
fof(conj_thm_2Ebool_2Eliteral__case__RAND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(A_27b,A_27c))
=> ! [V1N] :
( mem(V1N,arr(A_27a,A_27b))
=> ! [V2M] :
( mem(V2M,A_27a)
=> ap(V0P,ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),f53(A_27b,A_27a,V1N)),V2M)) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27c),f66(A_27b,A_27c,A_27a,V1N,V0P)),V2M) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2Eliteral__case__RATOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0N] :
( mem(V0N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1M] :
( mem(V1M,A_27a)
=> ! [V2b] :
( mem(V2b,A_27b)
=> ap(ap(ap(c_2Ebool_2Eliteral__case(A_27a,arr(A_27b,A_27c)),f55(A_27c,A_27b,A_27a,V0N)),V1M),V2b) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27c),f56(A_27b,A_27c,A_27a,V0N,V2b)),V1M) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2Eliteral__case__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2M] :
( mem(V2M,A_27a)
=> ! [V3N] :
( mem(V3N,A_27a)
=> ( ( V2M = V3N
& ! [V4x] :
( mem(V4x,A_27a)
=> ( V4x = V3N
=> ap(V0f,V4x) = ap(V1g,V4x) ) ) )
=> ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V0f),V2M) = ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),V1g),V3N) ) ) ) ) ) ) ) ).
fof(lameq_f67,axiom,
! [A_27b,A_27a,V1t] :
( mem(V1t,A_27b)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V2u] :
( mem(V2u,A_27b)
=> ! [V3x] : ap(f67(A_27b,A_27a,V1t,V0a,V2u),V3x) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(A_27a),V3x),V0a)),V1t),V2u) ) ) ) ).
fof(conj_thm_2Ebool_2Eliteral__case__id,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1t] :
( mem(V1t,A_27b)
=> ! [V2u] :
( mem(V2u,A_27b)
=> ap(ap(c_2Ebool_2Eliteral__case(A_27a,A_27b),f67(A_27b,A_27a,V1t,V0a,V2u)),V0a) = V1t ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EBOUNDED__THM,axiom,
! [V0v] :
( mem(V0v,bool)
=> ( p(ap(c_2Ebool_2EBOUNDED,V0v))
<=> $true ) ) ).
fof(conj_thm_2Ebool_2ELCOMM__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ( ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ! [V3z] :
( mem(V3z,A_27a)
=> ap(ap(V0f,V1x),ap(ap(V0f,V2y),V3z)) = ap(ap(V0f,ap(ap(V0f,V1x),V2y)),V3z) ) ) )
=> ( ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27a)
=> ap(ap(V0f,V4x),V5y) = ap(ap(V0f,V5y),V4x) ) )
=> ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7y] :
( mem(V7y,A_27a)
=> ! [V8z] :
( mem(V8z,A_27a)
=> ap(ap(V0f,V6x),ap(ap(V0f,V7y),V8z)) = ap(ap(V0f,V7y),ap(ap(V0f,V6x),V8z)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDATATYPE__TAG__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ( p(ap(c_2Ebool_2EDATATYPE(A_27a),V0x))
<=> $true ) ) ) ).
fof(conj_thm_2Ebool_2EDATATYPE__BOOL,axiom,
! [V0bool] :
( mem(V0bool,arr(bool,arr(bool,bool)))
=> ( p(ap(c_2Ebool_2EDATATYPE(bool),ap(ap(V0bool,c_2Ebool_2ET),c_2Ebool_2EF)))
<=> $true ) ) ).
fof(ax_thm_2Ebool_2Eitself__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ? [V0rep] :
( mem(V0rep,arr(ty_2Ebool_2Eitself(A_27a),A_27a))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(A_27a,ty_2Ebool_2Eitself(A_27a)),ap(c_2Emin_2E_3D(A_27a),c_2Ebool_2EARB(A_27a))),V0rep)) ) ) ).
fof(conj_thm_2Ebool_2EITSELF__UNIQUE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0i] :
( mem(V0i,ty_2Ebool_2Eitself(A_27a))
=> V0i = c_2Ebool_2Ethe__value(A_27a) ) ) ).
fof(conj_thm_2Ebool_2Eitself__Axiom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0e] :
( mem(V0e,A_27b)
=> ? [V1f] :
( mem(V1f,arr(ty_2Ebool_2Eitself(A_27a),A_27b))
& ap(V1f,c_2Ebool_2Ethe__value(A_27a)) = V0e ) ) ) ) ).
fof(conj_thm_2Ebool_2Eitself__induction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
=> ( p(ap(V0P,c_2Ebool_2Ethe__value(A_27a)))
=> ! [V1i] :
( mem(V1i,ty_2Ebool_2Eitself(A_27a))
=> p(ap(V0P,V1i)) ) ) ) ) ).
fof(ax_thm_2Ebool_2Eitself__case__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0b] :
( mem(V0b,A_27b)
=> ap(ap(c_2Ebool_2Eitself__case(A_27a,A_27b),c_2Ebool_2Ethe__value(A_27a)),V0b) = V0b ) ) ) ).
fof(conj_thm_2Ebool_2EFORALL__itself,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
=> ( ! [V1x] :
( mem(V1x,ty_2Ebool_2Eitself(A_27a))
=> p(ap(V0P,V1x)) )
<=> p(ap(V0P,c_2Ebool_2Ethe__value(A_27a))) ) ) ) ).
fof(conj_thm_2Ebool_2EEXISTS__itself,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Ebool_2Eitself(A_27a),bool))
=> ( ? [V1x] :
( mem(V1x,ty_2Ebool_2Eitself(A_27a))
& p(ap(V0P,V1x)) )
<=> p(ap(V0P,c_2Ebool_2Ethe__value(A_27a))) ) ) ) ).
fof(conj_thm_2Ebool_2EPULL__EXISTS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
=> p(V1Q) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(V0P,V3x))
=> p(V1Q) ) ) )
& ( ( ? [V4x] :
( mem(V4x,A_27a)
& p(ap(V0P,V4x)) )
& p(V1Q) )
<=> ? [V5x] :
( mem(V5x,A_27a)
& p(ap(V0P,V5x))
& p(V1Q) ) )
& ( ( p(V1Q)
& ? [V6x] :
( mem(V6x,A_27a)
& p(ap(V0P,V6x)) ) )
<=> ? [V7x] :
( mem(V7x,A_27a)
& p(V1Q)
& p(ap(V0P,V7x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EPULL__FORALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ( p(V1Q)
=> ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V0P,V2x)) ) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(V1Q)
=> p(ap(V0P,V3x)) ) ) )
& ( ( ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(V0P,V4x)) )
& p(V1Q) )
<=> ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(V0P,V5x))
& p(V1Q) ) ) )
& ( ( p(V1Q)
& ! [V6x] :
( mem(V6x,A_27a)
=> p(ap(V0P,V6x)) ) )
<=> ! [V7x] :
( mem(V7x,A_27a)
=> ( p(V1Q)
& p(ap(V0P,V7x)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EPEIRCE,axiom,
! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ( p(V0P)
=> p(V1Q) )
=> p(V0P) )
=> p(V0P) ) ) ) ).
fof(conj_thm_2Ebool_2EJRH__INDUCT__UTIL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,A_27a)
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( V2x = V1t
=> p(ap(V0P,V2x)) ) )
=> p(ap(c_2Ebool_2E_3F(A_27a),V0P)) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__EQ__IMP,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
| p(V1B) )
<=> ( ~ p(V0A)
=> p(V1B) ) ) ) ) ).
%------------------------------------------------------------------------------