ITP001 Axioms: ITP071+5.ax
%------------------------------------------------------------------------------
% File : ITP071+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 : quotient_option+2.ax [Gau20]
% : HL4071+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 15 ( 0 unt; 0 def)
% Number of atoms : 133 ( 5 equ)
% Maximal formula atoms : 19 ( 8 avg)
% Number of connectives : 118 ( 0 ~; 0 |; 4 &)
% ( 8 <=>; 106 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 15 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 1 con; 0-4 aty)
% Number of variables : 88 ( 88 !; 0 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(conj_thm_2Equotient__option_2EOPTION__MAP__I,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27a),c_2Ecombin_2EI(A_27a)) = c_2Ecombin_2EI(ty_2Eoption_2Eoption(A_27a)) ) ).
fof(conj_thm_2Equotient__option_2EOPTION__REL__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ( p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R),c_2Eoption_2ENONE(A_27a)),c_2Eoption_2ENONE(A_27a)))
<=> $true )
& ( p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R),ap(c_2Eoption_2ESOME(A_27a),V1x)),c_2Eoption_2ENONE(A_27a)))
<=> $false )
& ( p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R),c_2Eoption_2ENONE(A_27a)),ap(c_2Eoption_2ESOME(A_27a),V2y)))
<=> $false )
& ( p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R),ap(c_2Eoption_2ESOME(A_27a),V1x)),ap(c_2Eoption_2ESOME(A_27a),V2y)))
<=> p(ap(ap(V0R,V1x),V2y)) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2EOPTION__REL__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ap(c_2Eoption_2EOPTREL(A_27a,A_27a),c_2Emin_2E_3D(A_27a)) = c_2Emin_2E_3D(ty_2Eoption_2Eoption(A_27a)) ) ).
fof(conj_thm_2Equotient__option_2EOPTION__EQUIV,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ( p(ap(c_2Equotient_2EEQUIV(A_27a),V0R))
=> p(ap(c_2Equotient_2EEQUIV(ty_2Eoption_2Eoption(A_27a)),ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R))) ) ) ) ).
fof(conj_thm_2Equotient__option_2EOPTION__QUOTIENT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> p(ap(ap(ap(c_2Equotient_2EQUOTIENT(ty_2Eoption_2Eoption(A_27a),ty_2Eoption_2Eoption(A_27b)),ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R)),ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V1abs)),ap(c_2Eoption_2EOPTION__MAP(A_27b,A_27a),V2rep))) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2ENONE__PRS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> c_2Eoption_2ENONE(A_27b) = ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V1abs),c_2Eoption_2ENONE(A_27a)) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2ENONE__RSP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R),c_2Eoption_2ENONE(A_27a)),c_2Eoption_2ENONE(A_27a))) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2ESOME__PRS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3x] :
( mem(V3x,A_27b)
=> ap(c_2Eoption_2ESOME(A_27b),V3x) = ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V1abs),ap(c_2Eoption_2ESOME(A_27a),ap(V2rep,V3x))) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2ESOME__RSP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( p(ap(ap(V0R,V3x),V4y))
=> p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R),ap(c_2Eoption_2ESOME(A_27a),V3x)),ap(c_2Eoption_2ESOME(A_27a),V4y))) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2EIS__SOME__PRS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3x] :
( mem(V3x,ty_2Eoption_2Eoption(A_27b))
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27b),V3x))
<=> p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(ap(c_2Eoption_2EOPTION__MAP(A_27b,A_27a),V2rep),V3x))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2EIS__SOME__RSP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3x] :
( mem(V3x,ty_2Eoption_2Eoption(A_27a))
=> ! [V4y] :
( mem(V4y,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R),V3x),V4y))
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),V3x))
<=> p(ap(c_2Eoption_2EIS__SOME(A_27a),V4y)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2EIS__NONE__PRS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3x] :
( mem(V3x,ty_2Eoption_2Eoption(A_27b))
=> ( p(ap(c_2Eoption_2EIS__NONE(A_27b),V3x))
<=> p(ap(c_2Eoption_2EIS__NONE(A_27a),ap(ap(c_2Eoption_2EOPTION__MAP(A_27b,A_27a),V2rep),V3x))) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2EIS__NONE__RSP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs] :
( mem(V1abs,arr(A_27a,A_27b))
=> ! [V2rep] :
( mem(V2rep,arr(A_27b,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27b),V0R),V1abs),V2rep))
=> ! [V3x] :
( mem(V3x,ty_2Eoption_2Eoption(A_27a))
=> ! [V4y] :
( mem(V4y,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R),V3x),V4y))
=> ( p(ap(c_2Eoption_2EIS__NONE(A_27a),V3x))
<=> p(ap(c_2Eoption_2EIS__NONE(A_27a),V4y)) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2EOPTION__MAP__PRS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6a] :
( mem(V6a,ty_2Eoption_2Eoption(A_27c))
=> ! [V7f] :
( mem(V7f,arr(A_27c,A_27d))
=> ap(ap(c_2Eoption_2EOPTION__MAP(A_27c,A_27d),V7f),V6a) = ap(ap(c_2Eoption_2EOPTION__MAP(A_27b,A_27d),V4abs2),ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,A_27d,A_27c,A_27b),V1abs1),V5rep2),V7f)),ap(ap(c_2Eoption_2EOPTION__MAP(A_27c,A_27a),V2rep1),V6a))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2EOPTION__MAP__RSP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> ! [V0R1] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6a1] :
( mem(V6a1,ty_2Eoption_2Eoption(A_27a))
=> ! [V7a2] :
( mem(V7a2,ty_2Eoption_2Eoption(A_27a))
=> ! [V8f1] :
( mem(V8f1,arr(A_27a,A_27b))
=> ! [V9f2] :
( mem(V9f2,arr(A_27a,A_27b))
=> ( ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,A_27b),V0R1),V3R2),V8f1),V9f2))
& p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27a),V0R1),V6a1),V7a2)) )
=> p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27b,A_27b),V3R2),ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V8f1),V6a1)),ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V9f2),V7a2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------