ITP001 Axioms: ITP008+5.ax
%------------------------------------------------------------------------------
% File : ITP008+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 : combin+2.ax [Gau20]
% : HL4008+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 87 ( 8 unt; 0 def)
% Number of atoms : 417 ( 80 equ)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 335 ( 5 ~; 1 |; 9 &)
% ( 13 <=>; 307 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 49 ( 49 usr; 7 con; 0-5 aty)
% Number of variables : 371 ( 371 !; 0 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Ecombin_2E_3A_3E,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecombin_2E_3A_3E(A_27a,A_27b),arr(A_27b,arr(arr(A_27b,A_27a),A_27a))) ) ) ).
fof(mem_c_2Ecombin_2EASSOC,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ecombin_2EASSOC(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),bool)) ) ).
fof(mem_c_2Ecombin_2EC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Ecombin_2EC(A_27a,A_27b,A_27c),arr(arr(A_27a,arr(A_27b,A_27c)),arr(A_27b,arr(A_27a,A_27c)))) ) ) ) ).
fof(mem_c_2Ecombin_2ECOMM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecombin_2ECOMM(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,A_27b)),bool)) ) ) ).
fof(mem_c_2Ecombin_2EFAIL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecombin_2EFAIL(A_27a,A_27b),arr(A_27a,arr(A_27b,A_27a))) ) ) ).
fof(mem_c_2Ecombin_2EFCOMM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Ecombin_2EFCOMM(A_27a,A_27b,A_27c),arr(arr(A_27a,arr(A_27b,A_27a)),arr(arr(A_27c,arr(A_27a,A_27a)),bool))) ) ) ) ).
fof(mem_c_2Ecombin_2EI,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ecombin_2EI(A_27a),arr(A_27a,A_27a)) ) ).
fof(mem_c_2Ecombin_2EK,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecombin_2EK(A_27a,A_27b),arr(A_27a,arr(A_27b,A_27a))) ) ) ).
fof(mem_c_2Ecombin_2ELEFT__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecombin_2ELEFT__ID(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,A_27b)),arr(A_27a,bool))) ) ) ).
fof(mem_c_2Ecombin_2EMONOID,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Ecombin_2EMONOID(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,bool))) ) ).
fof(mem_c_2Ecombin_2ERIGHT__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecombin_2ERIGHT__ID(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,A_27a)),arr(A_27b,bool))) ) ) ).
fof(mem_c_2Ecombin_2ES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Ecombin_2ES(A_27a,A_27b,A_27c),arr(arr(A_27a,arr(A_27b,A_27c)),arr(arr(A_27a,A_27b),arr(A_27a,A_27c)))) ) ) ) ).
fof(mem_c_2Ecombin_2EUPDATE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecombin_2EUPDATE(A_27a,A_27b),arr(A_27a,arr(A_27b,arr(arr(A_27a,A_27b),arr(A_27a,A_27b))))) ) ) ).
fof(mem_c_2Ecombin_2EW,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Ecombin_2EW(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,A_27b)),arr(A_27a,A_27b))) ) ) ).
fof(mem_c_2Ecombin_2Eo,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> mem(c_2Ecombin_2Eo(A_27a,A_27b,A_27c),arr(arr(A_27c,A_27b),arr(arr(A_27a,A_27c),arr(A_27a,A_27b)))) ) ) ) ).
fof(lameq_f68,axiom,
! [A_27b,A_27a,V0x] : ap(f68(A_27b,A_27a),V0x) = k(A_27b,V0x) ).
fof(ax_thm_2Ecombin_2EK__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Ecombin_2EK(A_27a,A_27b) = f68(A_27b,A_27a) ) ) ).
fof(lameq_f69,axiom,
! [A_27b,A_27c,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2x] : ap(f69(A_27b,A_27c,A_27a,V0f,V1g),V2x) = ap(ap(V0f,V2x),ap(V1g,V2x)) ) ) ).
fof(lameq_f70,axiom,
! [A_27c,A_27a,A_27b,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1g] : ap(f70(A_27c,A_27a,A_27b,V0f),V1g) = f69(A_27b,A_27c,A_27a,V0f,V1g) ) ).
fof(lameq_f71,axiom,
! [A_27c,A_27a,A_27b,V0f] : ap(f71(A_27c,A_27a,A_27b),V0f) = f70(A_27c,A_27a,A_27b,V0f) ).
fof(ax_thm_2Ecombin_2ES__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> c_2Ecombin_2ES(A_27a,A_27b,A_27c) = f71(A_27c,A_27a,A_27b) ) ) ) ).
fof(ax_thm_2Ecombin_2EI__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> c_2Ecombin_2EI(A_27a) = ap(ap(c_2Ecombin_2ES(A_27a,arr(A_27a,A_27a),A_27a),c_2Ecombin_2EK(A_27a,arr(A_27a,A_27a))),c_2Ecombin_2EK(A_27a,A_27a)) ) ).
fof(lameq_f72,axiom,
! [A_27b,A_27c,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1x] :
( mem(V1x,A_27b)
=> ! [V2y] : ap(f72(A_27b,A_27c,A_27a,V0f,V1x),V2y) = ap(ap(V0f,V2y),V1x) ) ) ).
fof(lameq_f73,axiom,
! [A_27a,A_27c,A_27b,V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1x] : ap(f73(A_27a,A_27c,A_27b,V0f),V1x) = f72(A_27b,A_27c,A_27a,V0f,V1x) ) ).
fof(lameq_f74,axiom,
! [A_27a,A_27c,A_27b,V0f] : ap(f74(A_27a,A_27c,A_27b),V0f) = f73(A_27a,A_27c,A_27b,V0f) ).
fof(ax_thm_2Ecombin_2EC__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> c_2Ecombin_2EC(A_27a,A_27b,A_27c) = f74(A_27a,A_27c,A_27b) ) ) ) ).
fof(lameq_f75,axiom,
! [A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27b)))
=> ! [V1x] : ap(f75(A_27b,A_27a,V0f),V1x) = ap(ap(V0f,V1x),V1x) ) ).
fof(lameq_f76,axiom,
! [A_27b,A_27a,V0f] : ap(f76(A_27b,A_27a),V0f) = f75(A_27b,A_27a,V0f) ).
fof(ax_thm_2Ecombin_2EW__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Ecombin_2EW(A_27a,A_27b) = f76(A_27b,A_27a) ) ) ).
fof(lameq_f77,axiom,
! [A_27c,A_27b,A_27a,V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ! [V0f] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V2x] : ap(f77(A_27c,A_27b,A_27a,V1g,V0f),V2x) = ap(V0f,ap(V1g,V2x)) ) ) ).
fof(ax_thm_2Ecombin_2Eo__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27c),V0f),V1g) = f77(A_27c,A_27b,A_27a,V1g,V0f) ) ) ) ) ) ).
fof(ax_thm_2Ecombin_2EAPP__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27b)
=> ! [V1f] :
( mem(V1f,arr(A_27b,A_27a))
=> ap(ap(c_2Ecombin_2E_3A_3E(A_27a,A_27b),V0x),V1f) = ap(V1f,V0x) ) ) ) ) ).
fof(lameq_f78,axiom,
! [A_27b,A_27a,V1b] :
( mem(V1b,A_27b)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3c] : ap(f78(A_27b,A_27a,V1b,V0a,V2f),V3c) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(A_27a),V0a),V3c)),V1b),ap(V2f,V3c)) ) ) ) ).
fof(lameq_f79,axiom,
! [A_27b,A_27a,V1b] :
( mem(V1b,A_27b)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V2f] : ap(f79(A_27b,A_27a,V1b,V0a),V2f) = f78(A_27b,A_27a,V1b,V0a,V2f) ) ) ).
fof(ax_thm_2Ecombin_2EUPDATE__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1b] :
( mem(V1b,A_27b)
=> ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V0a),V1b) = f79(A_27b,A_27a,V1b,V0a) ) ) ) ) ).
fof(conj_thm_2Ecombin_2Eo__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27c,A_27a))
=> ! [V2x] :
( mem(V2x,A_27c)
=> ap(ap(ap(c_2Ecombin_2Eo(A_27c,A_27b,A_27a),V0f),V1g),V2x) = ap(V0f,ap(V1g,V2x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2Eo__ASSOC,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_27a))
=> ! [V2h] :
( mem(V2h,arr(A_27d,A_27c))
=> ap(ap(c_2Ecombin_2Eo(A_27d,A_27b,A_27a),V0f),ap(ap(c_2Ecombin_2Eo(A_27d,A_27a,A_27c),V1g),V2h)) = ap(ap(c_2Ecombin_2Eo(A_27d,A_27b,A_27c),ap(ap(c_2Ecombin_2Eo(A_27c,A_27b,A_27a),V0f),V1g)),V2h) ) ) ) ) ) ) ) ).
fof(lameq_f80,axiom,
! [A_27c,A_27a,V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V2x] : ap(f80(A_27c,A_27a,V0f),V2x) = ap(V0f,V2x) ) ).
fof(lameq_f81,axiom,
! [A_27a,A_27c,A_27b,V1g] :
( mem(V1g,arr(A_27b,A_27a))
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V3x] : ap(f81(A_27a,A_27c,A_27b,V1g,V0f),V3x) = ap(V0f,ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Ecombin_2Eo__ABS__L,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1g] :
( mem(V1g,arr(A_27b,A_27a))
=> ap(ap(c_2Ecombin_2Eo(A_27b,A_27c,A_27a),f80(A_27c,A_27a,V0f)),V1g) = f81(A_27a,A_27c,A_27b,V1g,V0f) ) ) ) ) ) ).
fof(lameq_f82,axiom,
! [A_27c,A_27a,V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ! [V2x] : ap(f82(A_27c,A_27a,V1g),V2x) = ap(V1g,V2x) ) ).
fof(lameq_f83,axiom,
! [A_27c,A_27b,A_27a,V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ! [V0f] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V3x] : ap(f83(A_27c,A_27b,A_27a,V1g,V0f),V3x) = ap(V0f,ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Ecombin_2Eo__ABS__R,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27c),V0f),f82(A_27c,A_27a,V1g)) = f83(A_27c,A_27b,A_27a,V1g,V0f) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EK__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ap(ap(c_2Ecombin_2EK(A_27a,A_27b),V0x),V1y) = V0x ) ) ) ) ).
fof(conj_thm_2Ecombin_2ES__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,A_27b))
=> ! [V2x] :
( mem(V2x,A_27a)
=> ap(ap(ap(c_2Ecombin_2ES(A_27a,A_27b,A_27c),V0f),V1g),V2x) = ap(ap(V0f,V2x),ap(V1g,V2x)) ) ) ) ) ) ) ).
fof(lameq_f84,axiom,
! [A_27b,A_27c,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V2x] : ap(f84(A_27b,A_27c,A_27a,V0f),V2x) = ap(V0f,V2x) ) ).
fof(lameq_f85,axiom,
! [A_27c,A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ! [V3x] : ap(f85(A_27c,A_27b,A_27a,V0f,V1g),V3x) = ap(ap(V0f,V3x),ap(V1g,V3x)) ) ) ).
fof(conj_thm_2Ecombin_2ES__ABS__L,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ap(ap(c_2Ecombin_2ES(A_27a,A_27c,A_27b),f84(A_27b,A_27c,A_27a,V0f)),V1g) = f85(A_27c,A_27b,A_27a,V0f,V1g) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2ES__ABS__R,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,A_27c))
=> ap(ap(c_2Ecombin_2ES(A_27a,A_27c,A_27b),V0f),f82(A_27c,A_27a,V1g)) = f85(A_27c,A_27b,A_27a,V0f,V1g) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EC__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)))
=> ! [V1x] :
( mem(V1x,A_27b)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ap(ap(ap(c_2Ecombin_2EC(A_27a,A_27b,A_27c),V0f),V1x),V2y) = ap(ap(V0f,V2y),V1x) ) ) ) ) ) ) ).
fof(lameq_f86,axiom,
! [A_27c,A_27b,A_27a,V0f] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1y] :
( mem(V1y,A_27c)
=> ! [V3x] : ap(f86(A_27c,A_27b,A_27a,V0f,V1y),V3x) = ap(ap(V0f,V3x),V1y) ) ) ).
fof(conj_thm_2Ecombin_2EC__ABS__L,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1y] :
( mem(V1y,A_27c)
=> ap(ap(c_2Ecombin_2EC(A_27a,A_27c,A_27b),f84(A_27b,A_27c,A_27a,V0f)),V1y) = f86(A_27c,A_27b,A_27a,V0f,V1y) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EW__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27b)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ap(ap(c_2Ecombin_2EW(A_27a,A_27b),V0f),V1x) = ap(ap(V0f,V1x),V1x) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EI__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ap(c_2Ecombin_2EI(A_27a),V0x) = V0x ) ) ).
fof(conj_thm_2Ecombin_2EI__o__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27b),c_2Ecombin_2EI(A_27b)),V0f) = V0f
& ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27a),V0f),c_2Ecombin_2EI(A_27a)) = V0f ) ) ) ) ).
fof(conj_thm_2Ecombin_2EK__o__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [A_27e] :
( ne(A_27e)
=> ! [A_27f] :
( ne(A_27f)
=> ( ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1v] :
( mem(V1v,A_27c)
=> ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),ap(c_2Ecombin_2EK(A_27c,A_27b),V1v)),V0f) = ap(c_2Ecombin_2EK(A_27c,A_27a),V1v) ) )
& ! [V2f] :
( mem(V2f,arr(A_27d,A_27e))
=> ! [V3v] :
( mem(V3v,A_27d)
=> ap(ap(c_2Ecombin_2Eo(A_27f,A_27e,A_27d),V2f),ap(c_2Ecombin_2EK(A_27d,A_27f),V3v)) = ap(c_2Ecombin_2EK(A_27e,A_27f),ap(V2f,V3v)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EUPDATE__APPLY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ( ! [V0a] :
( mem(V0a,A_27a)
=> ! [V1x] :
( mem(V1x,A_27b)
=> ! [V2f] :
( mem(V2f,arr(A_27a,A_27b))
=> ap(ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V0a),V1x),V2f),V0a) = V1x ) ) )
& ! [V3a] :
( mem(V3a,A_27c)
=> ! [V4b] :
( mem(V4b,A_27c)
=> ! [V5x] :
( mem(V5x,A_27d)
=> ! [V6f] :
( mem(V6f,arr(A_27c,A_27d))
=> ( V3a != V4b
=> ap(ap(ap(ap(c_2Ecombin_2EUPDATE(A_27c,A_27d),V3a),V5x),V6f),V4b) = ap(V6f,V4b) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EAPPLY__UPDATE__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2b] :
( mem(V2b,A_27b)
=> ! [V3c] :
( mem(V3c,A_27a)
=> ap(ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V2b),V0f),V3c) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(A_27a),V1a),V3c)),V2b),ap(V0f,V3c)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EUPDATE__COMMUTES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2b] :
( mem(V2b,A_27a)
=> ! [V3c] :
( mem(V3c,A_27b)
=> ! [V4d] :
( mem(V4d,A_27b)
=> ( V1a != V2b
=> ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2b),V4d),V0f)) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2b),V4d),ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EUPDATE__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2b] :
( mem(V2b,A_27b)
=> ! [V3c] :
( mem(V3c,A_27b)
=> ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V2b),V0f)) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EUPDATE__APPLY__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2b] :
( mem(V2b,A_27b)
=> ( ap(V0f,V1a) = V2b
<=> ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V2b),V0f) = V0f ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EUPDATE__APPLY__IMP__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1b] :
( mem(V1b,A_27b)
=> ! [V2a] :
( mem(V2a,A_27a)
=> ( ap(V0f,V2a) = V1b
=> ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V1b),V0f) = V0f ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EAPPLY__UPDATE__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),ap(V0f,V1a)),V0f) = V0f ) ) ) ) ).
fof(conj_thm_2Ecombin_2EUPD11__SAME__BASE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2b] :
( mem(V2b,A_27a)
=> ! [V3c] :
( mem(V3c,A_27b)
=> ! [V4d] :
( mem(V4d,A_27b)
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2b),V4d),V0f)
<=> ( ( V1a = V2b
& V3c = V4d )
| ( V1a != V2b
& ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f) = V0f
& ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2b),V4d),V0f) = V0f ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2ESAME__KEY__UPDATE__DIFFER,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_27c,A_27d))
=> ! [V1f1] :
( mem(V1f1,A_27a)
=> ! [V2f2] :
( mem(V2f2,A_27b)
=> ! [V3a] :
( mem(V3a,A_27c)
=> ! [V4b] :
( mem(V4b,A_27d)
=> ! [V5c] :
( mem(V5c,A_27d)
=> ( V4b != V5c
=> ap(ap(ap(c_2Ecombin_2EUPDATE(A_27c,A_27d),V3a),V4b),V0f) != ap(ap(ap(c_2Ecombin_2EUPDATE(A_27c,A_27d),V3a),V5c),V0f) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EUPD11__SAME__KEY__AND__BASE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2b] :
( mem(V2b,A_27b)
=> ! [V3c] :
( mem(V3c,A_27b)
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V2b),V0f) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f)
<=> V2b = V3c ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EUPD__SAME__KEY__UNWIND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f1] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V1f2] :
( mem(V1f2,arr(A_27a,A_27b))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3b] :
( mem(V3b,A_27b)
=> ! [V4c] :
( mem(V4c,A_27b)
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V3b),V0f1) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V4c),V1f2)
=> ( V3b = V4c
& ! [V5v] :
( mem(V5v,A_27b)
=> ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V5v),V0f1) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V5v),V1f2) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EGEN__LET__RAND,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0P] :
( mem(V0P,arr(A_27b,A_27a))
=> ! [V1f] :
( mem(V1f,arr(A_27c,A_27b))
=> ! [V2v] :
( mem(V2v,A_27c)
=> ap(V0P,ap(ap(c_2Ebool_2ELET(A_27c,A_27b),V1f),V2v)) = ap(ap(c_2Ebool_2ELET(A_27c,A_27a),ap(ap(c_2Ecombin_2Eo(A_27c,A_27a,A_27b),V0P),V1f)),V2v) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EGEN__LET__RATOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27b,arr(A_27c,A_27a)))
=> ! [V1v] :
( mem(V1v,A_27b)
=> ! [V2x] :
( mem(V2x,A_27c)
=> ap(ap(ap(c_2Ebool_2ELET(A_27b,arr(A_27c,A_27a)),V0f),V1v),V2x) = ap(ap(c_2Ebool_2ELET(A_27b,A_27a),ap(ap(c_2Ecombin_2EC(A_27b,A_27c,A_27a),V0f),V2x)),V1v) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2ELET__FORALL__ELIM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,bool))
=> ! [V1v] :
( mem(V1v,A_27a)
=> ( p(ap(ap(c_2Ebool_2ELET(A_27a,bool),V0f),V1v))
<=> p(ap(c_2Ebool_2E_21(A_27a),ap(ap(c_2Ecombin_2ES(A_27a,bool,bool),ap(ap(c_2Ecombin_2Eo(A_27a,arr(bool,bool),bool),c_2Emin_2E_3D_3D_3E),ap(ap(c_2Ecombin_2Eo(A_27a,bool,bool),c_2Emarker_2EAbbrev),ap(ap(c_2Ecombin_2EC(A_27a,A_27a,bool),c_2Emin_2E_3D(A_27a)),V1v)))),V0f))) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EGEN__literal__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_27a))
=> ! [V1f] :
( mem(V1f,arr(A_27c,A_27b))
=> ! [V2v] :
( mem(V2v,A_27c)
=> ap(V0P,ap(ap(c_2Ebool_2Eliteral__case(A_27c,A_27b),V1f),V2v)) = ap(ap(c_2Ebool_2Eliteral__case(A_27c,A_27a),ap(ap(c_2Ecombin_2Eo(A_27c,A_27a,A_27b),V0P),V1f)),V2v) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EGEN__literal__case__RATOR,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [V0f] :
( mem(V0f,arr(A_27b,arr(A_27c,A_27a)))
=> ! [V1v] :
( mem(V1v,A_27b)
=> ! [V2x] :
( mem(V2x,A_27c)
=> ap(ap(ap(c_2Ebool_2Eliteral__case(A_27b,arr(A_27c,A_27a)),V0f),V1v),V2x) = ap(ap(c_2Ebool_2Eliteral__case(A_27b,A_27a),ap(ap(c_2Ecombin_2EC(A_27b,A_27c,A_27a),V0f),V2x)),V1v) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2Eliteral__case__FORALL__ELIM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,bool))
=> ! [V1v] :
( mem(V1v,A_27a)
=> ( p(ap(ap(c_2Ebool_2Eliteral__case(A_27a,bool),V0f),V1v))
<=> p(ap(c_2Ebool_2E_21(A_27a),ap(ap(c_2Ecombin_2ES(A_27a,bool,bool),ap(ap(c_2Ecombin_2Eo(A_27a,arr(bool,bool),bool),c_2Emin_2E_3D_3D_3E),ap(ap(c_2Ecombin_2Eo(A_27a,bool,bool),c_2Emarker_2EAbbrev),ap(ap(c_2Ecombin_2EC(A_27a,A_27a,bool),c_2Emin_2E_3D(A_27a)),V1v)))),V0f))) ) ) ) ) ).
fof(ax_thm_2Ecombin_2EASSOC__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Ecombin_2EASSOC(A_27a),V0f))
<=> ! [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) ) ) ) ) ) ) ).
fof(ax_thm_2Ecombin_2ECOMM__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27b)))
=> ( p(ap(c_2Ecombin_2ECOMM(A_27a,A_27b),V0f))
<=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ap(ap(V0f,V1x),V2y) = ap(ap(V0f,V2y),V1x) ) ) ) ) ) ) ).
fof(ax_thm_2Ecombin_2EFCOMM__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_27a)))
=> ! [V1g] :
( mem(V1g,arr(A_27c,arr(A_27a,A_27a)))
=> ( p(ap(ap(c_2Ecombin_2EFCOMM(A_27a,A_27b,A_27c),V0f),V1g))
<=> ! [V2x] :
( mem(V2x,A_27c)
=> ! [V3y] :
( mem(V3y,A_27a)
=> ! [V4z] :
( mem(V4z,A_27b)
=> ap(ap(V1g,V2x),ap(ap(V0f,V3y),V4z)) = ap(ap(V0f,ap(ap(V1g,V2x),V3y)),V4z) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Ecombin_2ERIGHT__ID__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27a)))
=> ! [V1e] :
( mem(V1e,A_27b)
=> ( p(ap(ap(c_2Ecombin_2ERIGHT__ID(A_27a,A_27b),V0f),V1e))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ap(ap(V0f,V2x),V1e) = V2x ) ) ) ) ) ) ).
fof(ax_thm_2Ecombin_2ELEFT__ID__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1e] :
( mem(V1e,A_27a)
=> ( p(ap(ap(c_2Ecombin_2ELEFT__ID(A_27a,A_27b),V0f),V1e))
<=> ! [V2x] :
( mem(V2x,A_27b)
=> ap(ap(V0f,V1e),V2x) = V2x ) ) ) ) ) ) ).
fof(ax_thm_2Ecombin_2EMONOID__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1e] :
( mem(V1e,A_27a)
=> ( p(ap(ap(c_2Ecombin_2EMONOID(A_27a),V0f),V1e))
<=> ( p(ap(c_2Ecombin_2EASSOC(A_27a),V0f))
& p(ap(ap(c_2Ecombin_2ERIGHT__ID(A_27a,A_27a),V0f),V1e))
& p(ap(ap(c_2Ecombin_2ELEFT__ID(A_27a,A_27a),V0f),V1e)) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EASSOC__CONJ,axiom,
p(ap(c_2Ecombin_2EASSOC(bool),c_2Ebool_2E_2F_5C)) ).
fof(conj_thm_2Ecombin_2EASSOC__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Ecombin_2EASSOC(A_27a),V0f))
<=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ! [V3z] :
( mem(V3z,A_27a)
=> ap(ap(V0f,ap(ap(V0f,V1x),V2y)),V3z) = ap(ap(V0f,V1x),ap(ap(V0f,V2y),V3z)) ) ) ) ) ) ) ).
fof(conj_thm_2Ecombin_2EASSOC__DISJ,axiom,
p(ap(c_2Ecombin_2EASSOC(bool),c_2Ebool_2E_5C_2F)) ).
fof(conj_thm_2Ecombin_2EFCOMM__ASSOC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(ap(c_2Ecombin_2EFCOMM(A_27a,A_27a,A_27a),V0f),V0f))
<=> p(ap(c_2Ecombin_2EASSOC(A_27a),V0f)) ) ) ) ).
fof(conj_thm_2Ecombin_2EMONOID__CONJ__T,axiom,
p(ap(ap(c_2Ecombin_2EMONOID(bool),c_2Ebool_2E_2F_5C),c_2Ebool_2ET)) ).
fof(conj_thm_2Ecombin_2EMONOID__DISJ__F,axiom,
p(ap(ap(c_2Ecombin_2EMONOID(bool),c_2Ebool_2E_5C_2F),c_2Ebool_2EF)) ).
fof(ax_thm_2Ecombin_2EFAIL__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Ecombin_2EFAIL(A_27a,A_27b) = f68(A_27b,A_27a) ) ) ).
fof(conj_thm_2Ecombin_2EFAIL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27b)
=> ap(ap(c_2Ecombin_2EFAIL(A_27a,A_27b),V0x),V1y) = V0x ) ) ) ) ).
%------------------------------------------------------------------------------