ITP001 Axioms: ITP014+5.ax
%------------------------------------------------------------------------------
% File : ITP014+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 : pair+2.ax [Gau20]
% : HL4014+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 119 ( 2 unt; 0 def)
% Number of atoms : 690 ( 109 equ)
% Maximal formula atoms : 31 ( 5 avg)
% Number of connectives : 571 ( 0 ~; 2 |; 44 &)
% ( 29 <=>; 496 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 11 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 72 ( 72 usr; 3 con; 0-7 aty)
% Number of variables : 626 ( 607 !; 19 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Epair_2Eprod,axiom,
! [A0] :
( ne(A0)
=> ! [A1] :
( ne(A1)
=> ne(ty_2Epair_2Eprod(A0,A1)) ) ) ).
fof(mem_c_2Epair_2E_23_23,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> mem(c_2Epair_2E_23_23(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,A_27c),arr(arr(A_27b,A_27d),arr(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27c,A_27d))))) ) ) ) ) ).
fof(mem_c_2Epair_2E_2C,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2E_2C(A_27a,A_27b),arr(A_27a,arr(A_27b,ty_2Epair_2Eprod(A_27a,A_27b)))) ) ) ).
fof(mem_c_2Epair_2EABS__prod,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2EABS__prod(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),ty_2Epair_2Eprod(A_27a,A_27b))) ) ) ).
fof(mem_c_2Epair_2ECURRY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),arr(arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c),arr(A_27a,arr(A_27b,A_27c)))) ) ) ) ).
fof(mem_c_2Epair_2EFST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2EFST(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27a)) ) ) ).
fof(mem_c_2Epair_2ELEX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2ELEX(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27b,arr(A_27b,bool)),arr(ty_2Epair_2Eprod(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))))) ) ) ).
fof(mem_c_2Epair_2EPROD__ALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2EPROD__ALL(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_2Epair_2EREP__prod,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2EREP__prod(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27a,A_27b),arr(A_27a,arr(A_27b,bool)))) ) ) ).
fof(mem_c_2Epair_2ERPROD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2ERPROD(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27b,arr(A_27b,bool)),arr(ty_2Epair_2Eprod(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))))) ) ) ).
fof(mem_c_2Epair_2ESND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2ESND(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27b)) ) ) ).
fof(mem_c_2Epair_2ESWAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epair_2ESWAP(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27b,A_27a),ty_2Epair_2Eprod(A_27a,A_27b))) ) ) ).
fof(mem_c_2Epair_2EUNCURRY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),arr(arr(A_27a,arr(A_27b,A_27c)),arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))) ) ) ) ).
fof(mem_c_2Epair_2Epair__CASE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Epair_2Epair__CASE(A_27a,A_27b,A_27c),arr(ty_2Epair_2Eprod(A_27b,A_27c),arr(arr(A_27b,arr(A_27c,A_27a)),A_27a))) ) ) ) ).
fof(lameq_f111,axiom,
! [A_27a,A_27b,V4a] :
( mem(V4a,A_27a)
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V5b] : ap(f111(A_27a,A_27b,V4a,V2x,V3y),V5b) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(A_27a),V4a),V2x)),ap(ap(c_2Emin_2E_3D(A_27b),V5b),V3y)) ) ) ) ).
fof(lameq_f112,axiom,
! [A_27b,A_27a,V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ! [V4a] : ap(f112(A_27b,A_27a,V2x,V3y),V4a) = f111(A_27a,A_27b,V4a,V2x,V3y) ) ) ).
fof(lameq_f113,axiom,
! [A_27a,A_27b,V2x] :
( mem(V2x,A_27a)
=> ! [V1p] :
( mem(V1p,arr(A_27a,arr(A_27b,bool)))
=> ! [V3y] : ap(f113(A_27a,A_27b,V2x,V1p),V3y) = ap(ap(c_2Emin_2E_3D(arr(A_27a,arr(A_27b,bool))),V1p),f112(A_27b,A_27a,V2x,V3y)) ) ) ).
fof(lameq_f114,axiom,
! [A_27b,A_27a,V1p] :
( mem(V1p,arr(A_27a,arr(A_27b,bool)))
=> ! [V2x] : ap(f114(A_27b,A_27a,V1p),V2x) = ap(c_2Ebool_2E_3F(A_27b),f113(A_27a,A_27b,V2x,V1p)) ) ).
fof(lameq_f115,axiom,
! [A_27b,A_27a,V1p] : ap(f115(A_27b,A_27a),V1p) = ap(c_2Ebool_2E_3F(A_27a),f114(A_27b,A_27a,V1p)) ).
fof(ax_thm_2Epair_2Eprod__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ? [V0rep] :
( mem(V0rep,arr(ty_2Epair_2Eprod(A_27a,A_27b),arr(A_27a,arr(A_27b,bool))))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(arr(A_27a,arr(A_27b,bool)),ty_2Epair_2Eprod(A_27a,A_27b)),f115(A_27b,A_27a)),V0rep)) ) ) ) ).
fof(lameq_f116,axiom,
! [A_27a,A_27b,V5a] :
( mem(V5a,A_27a)
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27b)
=> ! [V6b] : ap(f116(A_27a,A_27b,V5a,V3x,V4y),V6b) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(A_27a),V5a),V3x)),ap(ap(c_2Emin_2E_3D(A_27b),V6b),V4y)) ) ) ) ).
fof(lameq_f117,axiom,
! [A_27b,A_27a,V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27b)
=> ! [V5a] : ap(f117(A_27b,A_27a,V3x,V4y),V5a) = f116(A_27a,A_27b,V5a,V3x,V4y) ) ) ).
fof(lameq_f118,axiom,
! [A_27a,A_27b,V3x] :
( mem(V3x,A_27a)
=> ! [V2p] :
( mem(V2p,arr(A_27a,arr(A_27b,bool)))
=> ! [V4y] : ap(f118(A_27a,A_27b,V3x,V2p),V4y) = ap(ap(c_2Emin_2E_3D(arr(A_27a,arr(A_27b,bool))),V2p),f117(A_27b,A_27a,V3x,V4y)) ) ) ).
fof(lameq_f119,axiom,
! [A_27b,A_27a,V2p] :
( mem(V2p,arr(A_27a,arr(A_27b,bool)))
=> ! [V3x] : ap(f119(A_27b,A_27a,V2p),V3x) = ap(c_2Ebool_2E_3F(A_27b),f118(A_27a,A_27b,V3x,V2p)) ) ).
fof(lameq_f120,axiom,
! [A_27b,A_27a,V2p] : ap(f120(A_27b,A_27a),V2p) = ap(c_2Ebool_2E_3F(A_27a),f119(A_27b,A_27a,V2p)) ).
fof(ax_thm_2Epair_2EABS__REP__prod,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(A_27a,A_27b))
=> ap(c_2Epair_2EABS__prod(A_27a,A_27b),ap(c_2Epair_2EREP__prod(A_27a,A_27b),V0a)) = V0a )
& ! [V1r] :
( mem(V1r,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(f120(A_27b,A_27a),V1r))
<=> ap(c_2Epair_2EREP__prod(A_27a,A_27b),ap(c_2Epair_2EABS__prod(A_27a,A_27b),V1r)) = V1r ) ) ) ) ) ).
fof(lameq_f121,axiom,
! [A_27a,A_27b,V2a] :
( mem(V2a,A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ! [V3b] : ap(f121(A_27a,A_27b,V2a,V0x,V1y),V3b) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(A_27a),V2a),V0x)),ap(ap(c_2Emin_2E_3D(A_27b),V3b),V1y)) ) ) ) ).
fof(lameq_f122,axiom,
! [A_27b,A_27a,V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ! [V2a] : ap(f122(A_27b,A_27a,V0x,V1y),V2a) = f121(A_27a,A_27b,V2a,V0x,V1y) ) ) ).
fof(ax_thm_2Epair_2ECOMMA__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y) = ap(c_2Epair_2EABS__prod(A_27a,A_27b),f122(A_27b,A_27a,V0x,V1y)) ) ) ) ) ).
fof(conj_thm_2Epair_2EPAIR__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3b] :
( mem(V3b,A_27b)
=> ( ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2a),V3b)
<=> ( V0x = V2a
& V1y = V3b ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3b] :
( mem(V3b,A_27b)
=> ( ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2a),V3b)
<=> ( V0x = V2a
& V1y = V3b ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EABS__PAIR__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,ty_2Epair_2Eprod(A_27a,A_27b))
=> ? [V1q] :
( mem(V1q,A_27a)
& ? [V2r] :
( mem(V2r,A_27b)
& V0x = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1q),V2r) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Epair__CASES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,ty_2Epair_2Eprod(A_27a,A_27b))
=> ? [V1q] :
( mem(V1q,A_27a)
& ? [V2r] :
( mem(V2r,A_27b)
& V0x = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1q),V2r) ) ) ) ) ) ).
fof(ax_thm_2Epair_2EPAIR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,ty_2Epair_2Eprod(A_27a,A_27b))
=> ap(ap(c_2Epair_2E_2C(A_27a,A_27b),ap(c_2Epair_2EFST(A_27a,A_27b),V0x)),ap(c_2Epair_2ESND(A_27a,A_27b),V0x)) = V0x ) ) ) ).
fof(conj_thm_2Epair_2EFST,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ap(c_2Epair_2EFST(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y)) = V0x ) ) ) ) ).
fof(conj_thm_2Epair_2ESND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ap(c_2Epair_2ESND(A_27a,A_27b),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V0x),V1y)) = V1y ) ) ) ) ).
fof(conj_thm_2Epair_2EPAIR__FST__SND__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V1q] :
( mem(V1q,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( V0p = V1q
<=> ( ap(c_2Epair_2EFST(A_27a,A_27b),V0p) = ap(c_2Epair_2EFST(A_27a,A_27b),V1q)
& ap(c_2Epair_2ESND(A_27a,A_27b),V0p) = ap(c_2Epair_2ESND(A_27a,A_27b),V1q) ) ) ) ) ) ) ).
fof(ax_thm_2Epair_2ESWAP__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,ty_2Epair_2Eprod(A_27b,A_27a))
=> ap(c_2Epair_2ESWAP(A_27a,A_27b),V0a) = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),ap(c_2Epair_2ESND(A_27b,A_27a),V0a)),ap(c_2Epair_2EFST(A_27b,A_27a),V0a)) ) ) ) ).
fof(ax_thm_2Epair_2ECURRY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27b)
=> ap(ap(ap(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),V0f),V1x),V2y) = ap(V0f,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1x),V2y)) ) ) ) ) ) ) ).
fof(ax_thm_2Epair_2EUNCURRY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1v] :
( mem(V1v,ty_2Epair_2Eprod(A_27a,A_27b))
=> ap(ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V0f),V1v) = ap(ap(V0f,ap(c_2Epair_2EFST(A_27a,A_27b),V1v)),ap(c_2Epair_2ESND(A_27a,A_27b),V1v)) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EUNCURRY__VAR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1v] :
( mem(V1v,ty_2Epair_2Eprod(A_27a,A_27b))
=> ap(ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V0f),V1v) = ap(ap(V0f,ap(c_2Epair_2EFST(A_27a,A_27b),V1v)),ap(c_2Epair_2ESND(A_27a,A_27b),V1v)) ) ) ) ) ) ).
fof(lameq_f123,axiom,
! [A_27c,A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1x] : ap(f123(A_27c,A_27a,A_27b,V0f),V1x) = ap(ap(V0f,ap(c_2Epair_2EFST(A_27a,A_27b),V1x)),ap(c_2Epair_2ESND(A_27a,A_27b),V1x)) ) ).
fof(conj_thm_2Epair_2EELIM__UNCURRY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V0f) = f123(A_27c,A_27a,A_27b,V0f) ) ) ) ) ).
fof(conj_thm_2Epair_2EUNCURRY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27b)
=> ap(ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V0f),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1x),V2y)) = ap(ap(V0f,V1x),V2y) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2ECURRY__UNCURRY__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ap(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V0f)) = V0f ) ) ) ) ).
fof(conj_thm_2Epair_2EUNCURRY__CURRY__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),ap(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),V0f)) = V0f ) ) ) ) ).
fof(conj_thm_2Epair_2ECURRY__ONE__ONE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ! [V1g] :
( mem(V1g,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ( ap(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),V0f) = ap(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),V1g)
<=> V0f = V1g ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EUNCURRY__ONE__ONE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,arr(A_27b,A_27c)))
=> ( ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V0f) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V1g)
<=> V0f = V1g ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Epair__Axiom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ? [V1fn] :
( mem(V1fn,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
& ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ap(V1fn,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2x),V3y)) = ap(ap(V0f,V2x),V3y) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EUNCURRY__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f_27] :
( mem(V0f_27,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1f] :
( mem(V1f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V2M_27] :
( mem(V2M_27,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V3M] :
( mem(V3M,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( ( V3M = V2M_27
& ! [V4x] :
( mem(V4x,A_27a)
=> ! [V5y] :
( mem(V5y,A_27b)
=> ( V2M_27 = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V4x),V5y)
=> ap(ap(V1f,V4x),V5y) = ap(ap(V0f_27,V4x),V5y) ) ) ) )
=> ap(ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V1f),V3M) = ap(ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V0f_27),V2M_27) ) ) ) ) ) ) ) ) ).
fof(lameq_f124,axiom,
! [A_27c,A_27a,A_27b,V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ! [V1p] : ap(f124(A_27c,A_27a,A_27b,V0P),V1p) = ap(V0P,V1p) ) ).
fof(lameq_f125,axiom,
! [A_27a,A_27c,A_27b,V2p1] :
( mem(V2p1,A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ! [V3p2] : ap(f125(A_27a,A_27c,A_27b,V2p1,V0P),V3p2) = ap(V0P,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2p1),V3p2)) ) ) ).
fof(lameq_f126,axiom,
! [A_27b,A_27c,A_27a,V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ! [V2p1] : ap(f126(A_27b,A_27c,A_27a,V0P),V2p1) = f125(A_27a,A_27c,A_27b,V2p1,V0P) ) ).
fof(conj_thm_2Epair_2ELAMBDA__PROD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> f124(A_27c,A_27a,A_27b,V0P) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),f126(A_27b,A_27c,A_27a,V0P)) ) ) ) ) ).
fof(conj_thm_2Epair_2EEXISTS__PROD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ( ? [V1p] :
( mem(V1p,ty_2Epair_2Eprod(A_27a,A_27b))
& p(ap(V0P,V1p)) )
<=> ? [V2p__1] :
( mem(V2p__1,A_27a)
& ? [V3p__2] :
( mem(V3p__2,A_27b)
& p(ap(V0P,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2p__1),V3p__2))) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EFORALL__PROD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ( ! [V1p] :
( mem(V1p,ty_2Epair_2Eprod(A_27a,A_27b))
=> p(ap(V0P,V1p)) )
<=> ! [V2p__1] :
( mem(V2p__1,A_27a)
=> ! [V3p__2] :
( mem(V3p__2,A_27b)
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2p__1),V3p__2))) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Epair__induction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool))
=> ( ! [V1p__1] :
( mem(V1p__1,A_27a)
=> ! [V2p__2] :
( mem(V2p__2,A_27b)
=> p(ap(V0P,ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1p__1),V2p__2))) ) )
=> ! [V3p] :
( mem(V3p,ty_2Epair_2Eprod(A_27a,A_27b))
=> p(ap(V0P,V3p)) ) ) ) ) ) ).
fof(ax_thm_2Epair_2EPROD__ALL__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))
=> ! [V2p] :
( mem(V2p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epair_2EPROD__ALL(A_27a,A_27b),V0P),V1Q),V2p))
<=> ( p(ap(V0P,ap(c_2Epair_2EFST(A_27a,A_27b),V2p)))
& p(ap(V1Q,ap(c_2Epair_2ESND(A_27a,A_27b),V2p))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EPROD__ALL__THM,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)
=> ! [V3y] :
( mem(V3y,A_27b)
=> ( p(ap(ap(ap(c_2Epair_2EPROD__ALL(A_27a,A_27b),V0P),V1Q),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2x),V3y)))
<=> ( p(ap(V0P,V2x))
& p(ap(V1Q,V3y)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EPROD__ALL__MONO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1P_27] :
( mem(V1P_27,arr(A_27a,bool))
=> ! [V2Q] :
( mem(V2Q,arr(A_27b,bool))
=> ! [V3Q_27] :
( mem(V3Q_27,arr(A_27b,bool))
=> ! [V4p] :
( mem(V4p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( ( ! [V5x] :
( mem(V5x,A_27a)
=> ( p(ap(V0P,V5x))
=> p(ap(V1P_27,V5x)) ) )
& ! [V6y] :
( mem(V6y,A_27b)
=> ( p(ap(V2Q,V6y))
=> p(ap(V3Q_27,V6y)) ) ) )
=> ( p(ap(ap(ap(c_2Epair_2EPROD__ALL(A_27a,A_27b),V0P),V2Q),V4p))
=> p(ap(ap(ap(c_2Epair_2EPROD__ALL(A_27a,A_27b),V1P_27),V3Q_27),V4p)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EPROD__ALL__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V1p_27] :
( mem(V1p_27,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V2P] :
( mem(V2P,arr(A_27a,bool))
=> ! [V3P_27] :
( mem(V3P_27,arr(A_27a,bool))
=> ! [V4Q] :
( mem(V4Q,arr(A_27b,bool))
=> ! [V5Q_27] :
( mem(V5Q_27,arr(A_27b,bool))
=> ( ( V0p = V1p_27
& ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7y] :
( mem(V7y,A_27b)
=> ( V1p_27 = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V6x),V7y)
=> ( p(ap(V2P,V6x))
<=> p(ap(V3P_27,V6x)) ) ) ) )
& ! [V8x] :
( mem(V8x,A_27a)
=> ! [V9y] :
( mem(V9y,A_27b)
=> ( V1p_27 = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V8x),V9y)
=> ( p(ap(V4Q,V9y))
<=> p(ap(V5Q_27,V9y)) ) ) ) ) )
=> ( p(ap(ap(ap(c_2Epair_2EPROD__ALL(A_27a,A_27b),V2P),V4Q),V0p))
<=> p(ap(ap(ap(c_2Epair_2EPROD__ALL(A_27a,A_27b),V3P_27),V5Q_27),V1p_27)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EELIM__PEXISTS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( ? [V1p] :
( mem(V1p,ty_2Epair_2Eprod(A_27a,A_27b))
& p(ap(ap(V0P,ap(c_2Epair_2EFST(A_27a,A_27b),V1p)),ap(c_2Epair_2ESND(A_27a,A_27b),V1p))) )
<=> ? [V2p1] :
( mem(V2p1,A_27a)
& ? [V3p2] :
( mem(V3p2,A_27b)
& p(ap(ap(V0P,V2p1),V3p2)) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EELIM__PFORALL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( ! [V1p] :
( mem(V1p,ty_2Epair_2Eprod(A_27a,A_27b))
=> p(ap(ap(V0P,ap(c_2Epair_2EFST(A_27a,A_27b),V1p)),ap(c_2Epair_2ESND(A_27a,A_27b),V1p))) )
<=> ! [V2p1] :
( mem(V2p1,A_27a)
=> ! [V3p2] :
( mem(V3p2,A_27b)
=> p(ap(ap(V0P,V2p1),V3p2)) ) ) ) ) ) ) ).
fof(lameq_f127,axiom,
! [A_27a,A_27b,V3x] :
( mem(V3x,A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ! [V4y] : ap(f127(A_27a,A_27b,V3x,V0P),V4y) = ap(ap(V0P,V3x),V4y) ) ) ).
fof(lameq_f128,axiom,
! [A_27b,A_27a,V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ! [V3x] : ap(f128(A_27b,A_27a,V0P),V3x) = f127(A_27a,A_27b,V3x,V0P) ) ).
fof(conj_thm_2Epair_2EPFORALL__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)) ) )
<=> p(ap(c_2Ebool_2E_21(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f128(A_27b,A_27a,V0P)))) ) ) ) ) ).
fof(conj_thm_2Epair_2EPEXISTS__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)) ) )
<=> p(ap(c_2Ebool_2E_3F(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f128(A_27b,A_27a,V0P)))) ) ) ) ) ).
fof(lameq_f129,axiom,
! [A_27b,A_27a,V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x] : ap(f129(A_27b,A_27a,V0P),V1x) = ap(V0P,V1x) ) ).
fof(conj_thm_2Epair_2EELIM__PEXISTS__EVAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Ebool_2E_3F(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f129(A_27b,A_27a,V0P))))
<=> ? [V2x] :
( mem(V2x,A_27a)
& p(ap(c_2Ebool_2E_3F(A_27b),ap(V0P,V2x))) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EELIM__PFORALL__EVAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0P] :
( mem(V0P,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Ebool_2E_21(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f129(A_27b,A_27a,V0P))))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(c_2Ebool_2E_21(A_27b),ap(V0P,V2x))) ) ) ) ) ) ).
fof(ax_thm_2Epair_2EPAIR__MAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27d))
=> ! [V2p] :
( mem(V2p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ap(ap(ap(c_2Epair_2E_23_23(A_27a,A_27b,A_27c,A_27d),V0f),V1g),V2p) = ap(ap(c_2Epair_2E_2C(A_27c,A_27d),ap(V0f,ap(c_2Epair_2EFST(A_27a,A_27b),V2p))),ap(V1g,ap(c_2Epair_2ESND(A_27a,A_27b),V2p))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EPAIR__MAP__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27c,A_27d))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ! [V3y] :
( mem(V3y,A_27c)
=> ap(ap(ap(c_2Epair_2E_23_23(A_27a,A_27c,A_27b,A_27d),V0f),V1g),ap(ap(c_2Epair_2E_2C(A_27a,A_27c),V2x),V3y)) = ap(ap(c_2Epair_2E_2C(A_27b,A_27d),ap(V0f,V2x)),ap(V1g,V3y)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EFST__PAIR__MAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27c))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27d))
=> ap(c_2Epair_2EFST(A_27c,A_27d),ap(ap(ap(c_2Epair_2E_23_23(A_27a,A_27b,A_27c,A_27d),V1f),V2g),V0p)) = ap(V1f,ap(c_2Epair_2EFST(A_27a,A_27b),V0p)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2ESND__PAIR__MAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27c))
=> ! [V2g] :
( mem(V2g,arr(A_27b,A_27d))
=> ap(c_2Epair_2ESND(A_27c,A_27d),ap(ap(ap(c_2Epair_2E_23_23(A_27a,A_27b,A_27c,A_27d),V1f),V2g),V0p)) = ap(V2g,ap(c_2Epair_2ESND(A_27a,A_27b),V0p)) ) ) ) ) ) ) ) ).
fof(lameq_f130,axiom,
! [A_27c,A_27a,A_27b,V3x] :
( mem(V3x,A_27a)
=> ! [V2N] :
( mem(V2N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V4y] : ap(f130(A_27c,A_27a,A_27b,V3x,V2N),V4y) = ap(ap(V2N,V3x),V4y) ) ) ).
fof(lameq_f131,axiom,
! [A_27b,A_27c,A_27a,V2N] :
( mem(V2N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V3x] : ap(f131(A_27b,A_27c,A_27a,V2N),V3x) = f130(A_27c,A_27a,A_27b,V3x,V2N) ) ).
fof(lameq_f132,axiom,
! [A_27d,A_27c,A_27a,A_27b,V2N] :
( mem(V2N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V5x] :
( mem(V5x,A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27c,A_27d))
=> ! [V6y] : ap(f132(A_27d,A_27c,A_27a,A_27b,V2N,V5x,V0P),V6y) = ap(V0P,ap(ap(V2N,V5x),V6y)) ) ) ) ).
fof(lameq_f133,axiom,
! [A_27b,A_27c,A_27d,A_27a,V2N] :
( mem(V2N,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V0P] :
( mem(V0P,arr(A_27c,A_27d))
=> ! [V5x] : ap(f133(A_27b,A_27c,A_27d,A_27a,V2N,V0P),V5x) = f132(A_27d,A_27c,A_27a,A_27b,V2N,V5x,V0P) ) ) ).
fof(conj_thm_2Epair_2ELET2__RAND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0P] :
( mem(V0P,arr(A_27c,A_27d))
=> ! [V1M] :
( mem(V1M,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V2N] :
( mem(V2N,arr(A_27a,arr(A_27b,A_27c)))
=> ap(V0P,ap(ap(c_2Ebool_2ELET(ty_2Epair_2Eprod(A_27a,A_27b),A_27c),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),f131(A_27b,A_27c,A_27a,V2N))),V1M)) = ap(ap(c_2Ebool_2ELET(ty_2Epair_2Eprod(A_27a,A_27b),A_27d),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27d),f133(A_27b,A_27c,A_27d,A_27a,V2N,V0P))),V1M) ) ) ) ) ) ) ) ).
fof(lameq_f134,axiom,
! [A_27c,A_27b,A_27a1,A_27a2,V3x] :
( mem(V3x,A_27a1)
=> ! [V1N] :
( mem(V1N,arr(A_27a1,arr(A_27a2,arr(A_27b,A_27c))))
=> ! [V4y] : ap(f134(A_27c,A_27b,A_27a1,A_27a2,V3x,V1N),V4y) = ap(ap(V1N,V3x),V4y) ) ) ).
fof(lameq_f135,axiom,
! [A_27a2,A_27b,A_27c,A_27a1,V1N] :
( mem(V1N,arr(A_27a1,arr(A_27a2,arr(A_27b,A_27c))))
=> ! [V3x] : ap(f135(A_27a2,A_27b,A_27c,A_27a1,V1N),V3x) = f134(A_27c,A_27b,A_27a1,A_27a2,V3x,V1N) ) ).
fof(lameq_f136,axiom,
! [A_27b,A_27a1,A_27c,A_27a2,V1N] :
( mem(V1N,arr(A_27a1,arr(A_27a2,arr(A_27b,A_27c))))
=> ! [V5x] :
( mem(V5x,A_27a1)
=> ! [V2b] :
( mem(V2b,A_27b)
=> ! [V6y] : ap(f136(A_27b,A_27a1,A_27c,A_27a2,V1N,V5x,V2b),V6y) = ap(ap(ap(V1N,V5x),V6y),V2b) ) ) ) ).
fof(lameq_f137,axiom,
! [A_27a2,A_27c,A_27b,A_27a1,V1N] :
( mem(V1N,arr(A_27a1,arr(A_27a2,arr(A_27b,A_27c))))
=> ! [V2b] :
( mem(V2b,A_27b)
=> ! [V5x] : ap(f137(A_27a2,A_27c,A_27b,A_27a1,V1N,V2b),V5x) = f136(A_27b,A_27a1,A_27c,A_27a2,V1N,V5x,V2b) ) ) ).
fof(conj_thm_2Epair_2ELET2__RATOR,axiom,
! [A_27a1] :
( ne(A_27a1)
=> ! [A_27a2] :
( ne(A_27a2)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0M] :
( mem(V0M,ty_2Epair_2Eprod(A_27a1,A_27a2))
=> ! [V1N] :
( mem(V1N,arr(A_27a1,arr(A_27a2,arr(A_27b,A_27c))))
=> ! [V2b] :
( mem(V2b,A_27b)
=> ap(ap(ap(c_2Ebool_2ELET(ty_2Epair_2Eprod(A_27a1,A_27a2),arr(A_27b,A_27c)),ap(c_2Epair_2EUNCURRY(A_27a1,A_27a2,arr(A_27b,A_27c)),f135(A_27a2,A_27b,A_27c,A_27a1,V1N))),V0M),V2b) = ap(ap(c_2Ebool_2ELET(ty_2Epair_2Eprod(A_27a1,A_27a2),A_27c),ap(c_2Epair_2EUNCURRY(A_27a1,A_27a2,A_27c),f137(A_27a2,A_27c,A_27b,A_27a1,V1N,V2b))),V0M) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Eo__UNCURRY__R,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27d,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27a,arr(A_27b,A_27d)))
=> ap(ap(c_2Ecombin_2Eo(ty_2Epair_2Eprod(A_27a,A_27b),A_27c,A_27d),V0f),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27d),V1g)) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),ap(ap(c_2Ecombin_2Eo(A_27a,arr(A_27b,A_27c),arr(A_27b,A_27d)),ap(c_2Ecombin_2Eo(A_27b,A_27c,A_27d),V0f)),V1g)) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EC__UNCURRY__L,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,arr(A_27d,A_27c))))
=> ! [V1x] :
( mem(V1x,A_27d)
=> ap(ap(c_2Ecombin_2EC(ty_2Epair_2Eprod(A_27a,A_27b),A_27d,A_27c),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,arr(A_27d,A_27c)),V0f)),V1x) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),ap(ap(c_2Ecombin_2EC(A_27a,A_27d,arr(A_27b,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,arr(A_27d,arr(A_27b,A_27c)),arr(A_27b,arr(A_27d,A_27c))),c_2Ecombin_2EC(A_27b,A_27d,A_27c)),V0f)),V1x)) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2ES__UNCURRY__R,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),arr(A_27d,A_27c)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,arr(A_27b,A_27d)))
=> ap(ap(c_2Ecombin_2ES(ty_2Epair_2Eprod(A_27a,A_27b),A_27d,A_27c),V0f),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27d),V1g)) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),ap(ap(c_2Ecombin_2ES(A_27a,arr(A_27b,A_27d),arr(A_27b,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,arr(arr(A_27b,A_27d),arr(A_27b,A_27c)),arr(A_27b,arr(A_27d,A_27c))),c_2Ecombin_2ES(A_27b,A_27d,A_27c)),ap(ap(c_2Ecombin_2Eo(A_27a,arr(A_27b,arr(A_27d,A_27c)),arr(A_27b,ty_2Epair_2Eprod(A_27a,A_27b))),ap(c_2Ecombin_2Eo(A_27b,arr(A_27d,A_27c),ty_2Epair_2Eprod(A_27a,A_27b)),V0f)),c_2Epair_2E_2C(A_27a,A_27b)))),V1g)) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EFORALL__UNCURRY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Ebool_2E_21(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),V0f)))
<=> p(ap(c_2Ebool_2E_21(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,bool,arr(A_27b,bool)),c_2Ebool_2E_21(A_27b)),V0f))) ) ) ) ) ).
fof(lameq_f138,axiom,
! [A_27c,A_27b,A_27a,V0P] :
( mem(V0P,arr(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool))
=> ! [V1f] : ap(f138(A_27c,A_27b,A_27a,V0P),V1f) = ap(V0P,V1f) ) ).
fof(lameq_f139,axiom,
! [A_27b,A_27c,A_27a,V2p] :
( mem(V2p,ty_2Epair_2Eprod(arr(A_27a,A_27b),arr(A_27a,A_27c)))
=> ! [V3a] : ap(f139(A_27b,A_27c,A_27a,V2p),V3a) = ap(ap(c_2Epair_2E_2C(A_27b,A_27c),ap(ap(c_2Epair_2EFST(arr(A_27a,A_27b),arr(A_27a,A_27c)),V2p),V3a)),ap(ap(c_2Epair_2ESND(arr(A_27a,A_27b),arr(A_27a,A_27c)),V2p),V3a)) ) ).
fof(lameq_f140,axiom,
! [A_27b,A_27c,A_27a,V0P] :
( mem(V0P,arr(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool))
=> ! [V2p] : ap(f140(A_27b,A_27c,A_27a,V0P),V2p) = ap(V0P,f139(A_27b,A_27c,A_27a,V2p)) ) ).
fof(conj_thm_2Epair_2EPAIR__FUN__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c)),bool))
=> ( p(ap(c_2Ebool_2E_3F_21(arr(A_27a,ty_2Epair_2Eprod(A_27b,A_27c))),f138(A_27c,A_27b,A_27a,V0P)))
<=> p(ap(c_2Ebool_2E_3F_21(ty_2Epair_2Eprod(arr(A_27a,A_27b),arr(A_27a,A_27c))),f140(A_27b,A_27c,A_27a,V0P))) ) ) ) ) ) ).
fof(ax_thm_2Epair_2Epair__CASE__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(A_27b,A_27c))
=> ! [V1f] :
( mem(V1f,arr(A_27b,arr(A_27c,A_27a)))
=> ap(ap(c_2Epair_2Epair__CASE(A_27a,A_27b,A_27c),V0p),V1f) = ap(ap(V1f,ap(c_2Epair_2EFST(A_27b,A_27c),V0p)),ap(c_2Epair_2ESND(A_27b,A_27c),V0p)) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Epair__case__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0x] :
( mem(V0x,A_27b)
=> ! [V1y] :
( mem(V1y,A_27c)
=> ! [V2f] :
( mem(V2f,arr(A_27b,arr(A_27c,A_27a)))
=> ap(ap(c_2Epair_2Epair__CASE(A_27a,A_27b,A_27c),ap(ap(c_2Epair_2E_2C(A_27b,A_27c),V0x),V1y)),V2f) = ap(ap(V2f,V0x),V1y) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Epair__case__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0x] :
( mem(V0x,A_27b)
=> ! [V1y] :
( mem(V1y,A_27c)
=> ! [V2f] :
( mem(V2f,arr(A_27b,arr(A_27c,A_27a)))
=> ap(ap(c_2Epair_2Epair__CASE(A_27a,A_27b,A_27c),ap(ap(c_2Epair_2E_2C(A_27b,A_27c),V0x),V1y)),V2f) = ap(ap(V2f,V0x),V1y) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Epair__case__cong,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f_27] :
( mem(V0f_27,arr(A_27b,arr(A_27c,A_27a)))
=> ! [V1M] :
( mem(V1M,ty_2Epair_2Eprod(A_27b,A_27c))
=> ! [V2M_27] :
( mem(V2M_27,ty_2Epair_2Eprod(A_27b,A_27c))
=> ! [V3f] :
( mem(V3f,arr(A_27b,arr(A_27c,A_27a)))
=> ( ( V1M = V2M_27
& ! [V4x] :
( mem(V4x,A_27b)
=> ! [V5y] :
( mem(V5y,A_27c)
=> ( V2M_27 = ap(ap(c_2Epair_2E_2C(A_27b,A_27c),V4x),V5y)
=> ap(ap(V3f,V4x),V5y) = ap(ap(V0f_27,V4x),V5y) ) ) ) )
=> ap(ap(c_2Epair_2Epair__CASE(A_27a,A_27b,A_27c),V1M),V3f) = ap(ap(c_2Epair_2Epair__CASE(A_27a,A_27b,A_27c),V2M_27),V0f_27) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Epair__case__eq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(A_27b,A_27c))
=> ! [V1f] :
( mem(V1f,arr(A_27b,arr(A_27c,A_27a)))
=> ! [V2v] :
( mem(V2v,A_27a)
=> ( ap(ap(c_2Epair_2Epair__CASE(A_27a,A_27b,A_27c),V0p),V1f) = V2v
<=> ? [V3x] :
( mem(V3x,A_27b)
& ? [V4y] :
( mem(V4y,A_27c)
& V0p = ap(ap(c_2Epair_2E_2C(A_27b,A_27c),V3x),V4y)
& ap(ap(V1f,V3x),V4y) = V2v ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Edatatype__pair,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0pair] :
( mem(V0pair,arr(arr(A_27a,arr(A_27b,ty_2Epair_2Eprod(A_27a,A_27b))),A_27c))
=> p(ap(c_2Ebool_2EDATATYPE(A_27c),ap(V0pair,c_2Epair_2E_2C(A_27a,A_27b)))) ) ) ) ) ).
fof(lameq_f141,axiom,
! [A_27a,A_27b,V3t] :
( mem(V3t,A_27b)
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4u] :
( mem(V4u,A_27a)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V2s] :
( mem(V2s,A_27a)
=> ! [V5v] : ap(f141(A_27a,A_27b,V3t,V1R2,V4u,V0R1,V2s),V5v) = ap(ap(c_2Ebool_2E_5C_2F,ap(ap(V0R1,V2s),V4u)),ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(A_27a),V2s),V4u)),ap(ap(V1R2,V3t),V5v))) ) ) ) ) ) ).
fof(lameq_f142,axiom,
! [A_27b,A_27a,V3t] :
( mem(V3t,A_27b)
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V2s] :
( mem(V2s,A_27a)
=> ! [V4u] : ap(f142(A_27b,A_27a,V3t,V1R2,V0R1,V2s),V4u) = f141(A_27a,A_27b,V3t,V1R2,V4u,V0R1,V2s) ) ) ) ) ).
fof(lameq_f143,axiom,
! [A_27a,A_27b,V2s] :
( mem(V2s,A_27a)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V3t] : ap(f143(A_27a,A_27b,V2s,V0R1,V1R2),V3t) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f142(A_27b,A_27a,V3t,V1R2,V0R1,V2s)) ) ) ) ).
fof(lameq_f144,axiom,
! [A_27b,A_27a,V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V2s] : ap(f144(A_27b,A_27a,V0R1,V1R2),V2s) = f143(A_27a,A_27b,V2s,V0R1,V1R2) ) ) ).
fof(ax_thm_2Epair_2ELEX__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R1),V1R2) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool)),f144(A_27b,A_27a,V0R1,V1R2)) ) ) ) ) ).
fof(conj_thm_2Epair_2ELEX__DEF__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3b] :
( mem(V3b,A_27b)
=> ! [V4c] :
( mem(V4c,A_27a)
=> ! [V5d] :
( mem(V5d,A_27b)
=> ( p(ap(ap(ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R1),V1R2),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2a),V3b)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V4c),V5d)))
<=> ( p(ap(ap(V0R1,V2a),V4c))
| ( V2a = V4c
& p(ap(ap(V1R2,V3b),V5d)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2ELEX__MONO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27a,arr(A_27a,bool)))
=> ! [V2R3] :
( mem(V2R3,arr(A_27b,arr(A_27b,bool)))
=> ! [V3R4] :
( mem(V3R4,arr(A_27b,arr(A_27b,bool)))
=> ! [V4x] :
( mem(V4x,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V5y] :
( mem(V5y,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( ( ! [V6x] :
( mem(V6x,A_27a)
=> ! [V7y] :
( mem(V7y,A_27a)
=> ( p(ap(ap(V0R1,V6x),V7y))
=> p(ap(ap(V1R2,V6x),V7y)) ) ) )
& ! [V8x] :
( mem(V8x,A_27b)
=> ! [V9y] :
( mem(V9y,A_27b)
=> ( p(ap(ap(V2R3,V8x),V9y))
=> p(ap(ap(V3R4,V8x),V9y)) ) ) ) )
=> ( p(ap(ap(ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R1),V2R3),V4x),V5y))
=> p(ap(ap(ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V1R2),V3R4),V4x),V5y)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EWF__LEX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,arr(A_27b,bool)))
=> ( ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
& p(ap(c_2Erelation_2EWF(A_27b),V1Q)) )
=> p(ap(c_2Erelation_2EWF(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R),V1Q))) ) ) ) ) ) ).
fof(lameq_f145,axiom,
! [A_27a,A_27b,V2s] :
( mem(V2s,A_27a)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V4u] :
( mem(V4u,A_27a)
=> ! [V3t] :
( mem(V3t,A_27b)
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V5v] : ap(f145(A_27a,A_27b,V2s,V0R1,V4u,V3t,V1R2),V5v) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(V0R1,V2s),V4u)),ap(ap(V1R2,V3t),V5v)) ) ) ) ) ) ).
fof(lameq_f146,axiom,
! [A_27b,A_27a,V2s] :
( mem(V2s,A_27a)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V3t] :
( mem(V3t,A_27b)
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4u] : ap(f146(A_27b,A_27a,V2s,V0R1,V3t,V1R2),V4u) = f145(A_27a,A_27b,V2s,V0R1,V4u,V3t,V1R2) ) ) ) ) ).
fof(lameq_f147,axiom,
! [A_27a,A_27b,V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V2s] :
( mem(V2s,A_27a)
=> ! [V3t] : ap(f147(A_27a,A_27b,V1R2,V0R1,V2s),V3t) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,bool),f146(A_27b,A_27a,V2s,V0R1,V3t,V1R2)) ) ) ) ).
fof(lameq_f148,axiom,
! [A_27b,A_27a,V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V2s] : ap(f148(A_27b,A_27a,V1R2,V0R1),V2s) = f147(A_27a,A_27b,V1R2,V0R1,V2s) ) ) ).
fof(ax_thm_2Epair_2ERPROD__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ap(ap(c_2Epair_2ERPROD(A_27a,A_27b),V0R1),V1R2) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,arr(ty_2Epair_2Eprod(A_27a,A_27b),bool)),f148(A_27b,A_27a,V1R2,V0R1)) ) ) ) ) ).
fof(conj_thm_2Epair_2EWF__RPROD,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1Q] :
( mem(V1Q,arr(A_27b,arr(A_27b,bool)))
=> ( ( p(ap(c_2Erelation_2EWF(A_27a),V0R))
& p(ap(c_2Erelation_2EWF(A_27b),V1Q)) )
=> p(ap(c_2Erelation_2EWF(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2ERPROD(A_27a,A_27b),V0R),V1Q))) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Etotal__LEX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ( ( p(ap(c_2Erelation_2Etotal(A_27a),V0R1))
& p(ap(c_2Erelation_2Etotal(A_27b),V1R2)) )
=> p(ap(c_2Erelation_2Etotal(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R1),V1R2))) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Etransitive__LEX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ( ( p(ap(c_2Erelation_2Etransitive(A_27a),V0R1))
& p(ap(c_2Erelation_2Etransitive(A_27b),V1R2)) )
=> p(ap(c_2Erelation_2Etransitive(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R1),V1R2))) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Ereflexive__LEX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ( p(ap(c_2Erelation_2Ereflexive(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R1),V1R2)))
<=> ( p(ap(c_2Erelation_2Ereflexive(A_27a),V0R1))
| p(ap(c_2Erelation_2Ereflexive(A_27b),V1R2)) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2Esymmetric__LEX,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ( ( p(ap(c_2Erelation_2Esymmetric(A_27a),V0R1))
& p(ap(c_2Erelation_2Esymmetric(A_27b),V1R2)) )
=> p(ap(c_2Erelation_2Esymmetric(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R1),V1R2))) ) ) ) ) ) ).
fof(conj_thm_2Epair_2ELEX__CONG,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V2v1] :
( mem(V2v1,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V3v2] :
( mem(V3v2,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V4R1_27] :
( mem(V4R1_27,arr(A_27a,arr(A_27a,bool)))
=> ! [V5R2_27] :
( mem(V5R2_27,arr(A_27b,arr(A_27b,bool)))
=> ! [V6v1_27] :
( mem(V6v1_27,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V7v2_27] :
( mem(V7v2_27,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( ( V2v1 = V6v1_27
& V3v2 = V7v2_27
& ! [V8a] :
( mem(V8a,A_27a)
=> ! [V9b] :
( mem(V9b,A_27b)
=> ! [V10c] :
( mem(V10c,A_27a)
=> ! [V11d] :
( mem(V11d,A_27b)
=> ( ( V6v1_27 = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V8a),V9b)
& V7v2_27 = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V10c),V11d) )
=> ( p(ap(ap(V0R1,V8a),V10c))
<=> p(ap(ap(V4R1_27,V8a),V10c)) ) ) ) ) ) )
& ! [V12a] :
( mem(V12a,A_27a)
=> ! [V13b] :
( mem(V13b,A_27b)
=> ! [V14c] :
( mem(V14c,A_27a)
=> ! [V15d] :
( mem(V15d,A_27b)
=> ( ( V6v1_27 = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V12a),V13b)
& V7v2_27 = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V14c),V15d)
& V12a = V14c )
=> ( p(ap(ap(V1R2,V13b),V15d))
<=> p(ap(ap(V5R2_27,V13b),V15d)) ) ) ) ) ) ) )
=> ( p(ap(ap(ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V0R1),V1R2),V2v1),V3v2))
<=> p(ap(ap(ap(ap(c_2Epair_2ELEX(A_27a,A_27b),V4R1_27),V5R2_27),V6v1_27),V7v2_27)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2EFST__EQ__EQUIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( ap(c_2Epair_2EFST(A_27a,A_27b),V0p) = V1x
<=> ? [V2y] :
( mem(V2y,A_27b)
& V0p = ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V1x),V2y) ) ) ) ) ) ) ).
fof(conj_thm_2Epair_2ESND__EQ__EQUIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0p] :
( mem(V0p,ty_2Epair_2Eprod(A_27b,A_27a))
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( ap(c_2Epair_2ESND(A_27b,A_27a),V0p) = V1y
<=> ? [V2x] :
( mem(V2x,A_27b)
& V0p = ap(ap(c_2Epair_2E_2C(A_27b,A_27a),V2x),V1y) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------