TPTP Problem File: ITP011+2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP011+2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Equotient__option_2EOPTION__REL__def.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Equotient__option_2EOPTION__REL__def.p [Gau19]
% : HL405001+2.p [TPAP]
% Status : Theorem
% Rating : 0.78 v8.2.0, 0.81 v7.5.0
% Syntax : Number of formulae : 48 ( 11 unt; 0 def)
% Number of atoms : 247 ( 30 equ)
% Maximal formula atoms : 57 ( 5 avg)
% Number of connectives : 210 ( 11 ~; 7 |; 48 &)
% ( 42 <=>; 102 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 6 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 25 ( 25 usr; 8 con; 0-2 aty)
% Number of variables : 101 ( 97 !; 4 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
%------------------------------------------------------------------------------
fof(ne_ty_2Eoption_2Eoption,axiom,
! [A0] :
( ne(A0)
=> ne(ty_2Eoption_2Eoption(A0)) ) ).
fof(mem_c_2Eoption_2EOPTION__JOIN,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eoption_2EOPTION__JOIN(A_27a),arr(ty_2Eoption_2Eoption(ty_2Eoption_2Eoption(A_27a)),ty_2Eoption_2Eoption(A_27a))) ) ).
fof(mem_c_2Eoption_2EOPTION__MAP,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(ty_2Eoption_2Eoption(A_27a),ty_2Eoption_2Eoption(A_27b)))) ) ) ).
fof(mem_c_2Eoption_2Eoption__CASE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eoption_2Eoption__CASE(A_27a,A_27b),arr(ty_2Eoption_2Eoption(A_27a),arr(A_27b,arr(arr(A_27a,A_27b),A_27b)))) ) ) ).
fof(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem(c_2Emin_2E_3D_3D_3E,arr(bool,arr(bool,bool))) ).
fof(ax_imp_p,axiom,
! [Q] :
( mem(Q,bool)
=> ! [R] :
( mem(R,bool)
=> ( p(ap(ap(c_2Emin_2E_3D_3D_3E,Q),R))
<=> ( p(Q)
=> p(R) ) ) ) ) ).
fof(mem_c_2Eoption_2EIS__NONE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eoption_2EIS__NONE(A_27a),arr(ty_2Eoption_2Eoption(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_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
fof(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
fof(mem_c_2Eoption_2EIS__SOME,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eoption_2EIS__SOME(A_27a),arr(ty_2Eoption_2Eoption(A_27a),bool)) ) ).
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(mem_c_2Eoption_2ETHE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eoption_2ETHE(A_27a),arr(ty_2Eoption_2Eoption(A_27a),A_27a)) ) ).
fof(mem_c_2Eoption_2ESOME,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eoption_2ESOME(A_27a),arr(A_27a,ty_2Eoption_2Eoption(A_27a))) ) ).
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_2Eoption_2ENONE,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eoption_2ENONE(A_27a),ty_2Eoption_2Eoption(A_27a)) ) ).
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_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_2Eoption_2EOPTREL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eoption_2EOPTREL(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(ty_2Eoption_2Eoption(A_27a),arr(ty_2Eoption_2Eoption(A_27b),bool)))) ) ) ).
fof(mem_c_2Emin_2E_3D,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Emin_2E_3D(A_27a),arr(A_27a,arr(A_27a,bool))) ) ).
fof(ax_eq_p,axiom,
! [A] :
( ne(A)
=> ! [X] :
( mem(X,A)
=> ! [Y] :
( mem(Y,A)
=> ( p(ap(ap(c_2Emin_2E_3D(A),X),Y))
<=> X = Y ) ) ) ) ).
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(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(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__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_2ENOT__CLAUSES,axiom,
( ! [V0t] :
( mem(V0t,bool)
=> ( ~ ~ p(V0t)
<=> p(V0t) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
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__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_2Eoption_2Eoption__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0e] :
( mem(V0e,A_27b)
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2e] :
( mem(V2e,ty_2Eoption_2Eoption(A_27a))
=> ( ! [V3x] :
( mem(V3x,A_27a)
=> ! [V4y] :
( mem(V4y,A_27a)
=> ( ap(c_2Eoption_2ESOME(A_27a),V3x) = ap(c_2Eoption_2ESOME(A_27a),V4y)
<=> V3x = V4y ) ) )
& ! [V5x] :
( mem(V5x,A_27a)
=> ap(c_2Eoption_2ETHE(A_27a),ap(c_2Eoption_2ESOME(A_27a),V5x)) = V5x )
& ! [V6x] :
( mem(V6x,A_27a)
=> c_2Eoption_2ENONE(A_27a) != ap(c_2Eoption_2ESOME(A_27a),V6x) )
& ! [V7x] :
( mem(V7x,A_27a)
=> ap(c_2Eoption_2ESOME(A_27a),V7x) != c_2Eoption_2ENONE(A_27a) )
& ! [V8x] :
( mem(V8x,A_27a)
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(c_2Eoption_2ESOME(A_27a),V8x)))
<=> $true ) )
& ( p(ap(c_2Eoption_2EIS__SOME(A_27a),c_2Eoption_2ENONE(A_27a)))
<=> $false )
& ! [V9x] :
( mem(V9x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__NONE(A_27a),V9x))
<=> V9x = c_2Eoption_2ENONE(A_27a) ) )
& ! [V10x] :
( mem(V10x,ty_2Eoption_2Eoption(A_27a))
=> ( ~ p(ap(c_2Eoption_2EIS__SOME(A_27a),V10x))
<=> V10x = c_2Eoption_2ENONE(A_27a) ) )
& ! [V11x] :
( mem(V11x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),V11x))
=> ap(c_2Eoption_2ESOME(A_27a),ap(c_2Eoption_2ETHE(A_27a),V11x)) = V11x ) )
& ! [V12x] :
( mem(V12x,ty_2Eoption_2Eoption(A_27a))
=> ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,ty_2Eoption_2Eoption(A_27a)),V12x),c_2Eoption_2ENONE(A_27a)),c_2Eoption_2ESOME(A_27a)) = V12x )
& ! [V13x] :
( mem(V13x,ty_2Eoption_2Eoption(A_27a))
=> ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,ty_2Eoption_2Eoption(A_27a)),V13x),V13x),c_2Eoption_2ESOME(A_27a)) = V13x )
& ! [V14x] :
( mem(V14x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__NONE(A_27a),V14x))
=> ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),V14x),V0e),V1f) = V0e ) )
& ! [V15x] :
( mem(V15x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),V15x))
=> ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),V15x),V0e),V1f) = ap(V1f,ap(c_2Eoption_2ETHE(A_27a),V15x)) ) )
& ! [V16x] :
( mem(V16x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),V16x))
=> ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,ty_2Eoption_2Eoption(A_27a)),V16x),V2e),c_2Eoption_2ESOME(A_27a)) = V16x ) )
& ! [V17v] :
( mem(V17v,A_27b)
=> ! [V18f] :
( mem(V18f,arr(A_27a,A_27b))
=> ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),c_2Eoption_2ENONE(A_27a)),V17v),V18f) = V17v ) )
& ! [V19x] :
( mem(V19x,A_27a)
=> ! [V20v] :
( mem(V20v,A_27b)
=> ! [V21f] :
( mem(V21f,arr(A_27a,A_27b))
=> ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),ap(c_2Eoption_2ESOME(A_27a),V19x)),V20v),V21f) = ap(V21f,V19x) ) ) )
& ! [V22f] :
( mem(V22f,arr(A_27a,A_27b))
=> ! [V23x] :
( mem(V23x,A_27a)
=> ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V22f),ap(c_2Eoption_2ESOME(A_27a),V23x)) = ap(c_2Eoption_2ESOME(A_27b),ap(V22f,V23x)) ) )
& ! [V24f] :
( mem(V24f,arr(A_27a,A_27b))
=> ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V24f),c_2Eoption_2ENONE(A_27a)) = c_2Eoption_2ENONE(A_27b) )
& ap(c_2Eoption_2EOPTION__JOIN(A_27a),c_2Eoption_2ENONE(ty_2Eoption_2Eoption(A_27a))) = c_2Eoption_2ENONE(A_27a)
& ! [V25x] :
( mem(V25x,ty_2Eoption_2Eoption(A_27a))
=> ap(c_2Eoption_2EOPTION__JOIN(A_27a),ap(c_2Eoption_2ESOME(ty_2Eoption_2Eoption(A_27a)),V25x)) = V25x ) ) ) ) ) ) ) ).
fof(ax_thm_2Eoption_2EOPTREL__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0R] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x] :
( mem(V1x,ty_2Eoption_2Eoption(A_27a))
=> ! [V2y] :
( mem(V2y,ty_2Eoption_2Eoption(A_27b))
=> ( p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27b),V0R),V1x),V2y))
<=> ( ( V1x = c_2Eoption_2ENONE(A_27a)
& V2y = c_2Eoption_2ENONE(A_27b) )
| ? [V3x0] :
( mem(V3x0,A_27a)
& ? [V4y0] :
( mem(V4y0,A_27b)
& V1x = ap(c_2Eoption_2ESOME(A_27a),V3x0)
& V2y = ap(c_2Eoption_2ESOME(A_27b),V4y0)
& p(ap(ap(V0R,V3x0),V4y0)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient__option_2EOPTION__REL__def,conjecture,
! [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)) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------