TPTP Problem File: ITP009+2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP009+2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Equotient_2EFUN__REL__EQ__REL.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_2EFUN__REL__EQ__REL.p [Gau19]
% : HL404001+2.p [TPAP]
% Status : Theorem
% Rating : 0.92 v8.2.0, 0.94 v7.5.0
% Syntax : Number of formulae : 37 ( 10 unt; 0 def)
% Number of atoms : 171 ( 10 equ)
% Maximal formula atoms : 18 ( 4 avg)
% Number of connectives : 138 ( 4 ~; 0 |; 18 &)
% ( 23 <=>; 93 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 18 ( 18 usr; 7 con; 0-4 aty)
% Number of variables : 85 ( 85 !; 0 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
%------------------------------------------------------------------------------
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_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_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_2Equotient_2E_2D_2D_3E,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [A_27c] :
( ne(A_27c)
=> ! [A_27d] :
( ne(A_27d)
=> mem(c_2Equotient_2E_2D_2D_3E(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,A_27c),arr(arr(A_27b,A_27d),arr(arr(A_27c,A_27b),arr(A_27a,A_27d))))) ) ) ) ) ).
fof(mem_c_2Equotient_2E_3D_3D_3D_3E,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Equotient_2E_3D_3D_3D_3E(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27b,arr(A_27b,bool)),arr(arr(A_27a,A_27b),arr(arr(A_27a,A_27b),bool))))) ) ) ).
fof(mem_c_2Equotient_2EQUOTIENT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Equotient_2EQUOTIENT(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,bool)),arr(arr(A_27a,A_27b),arr(arr(A_27b,A_27a),bool)))) ) ) ).
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_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_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_2Equotient_2Erespects,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Equotient_2Erespects(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,A_27b)),arr(A_27a,A_27b))) ) ) ).
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(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_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_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_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_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_2Equotient_2EQUOTIENT__REL,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))
=> ! [V3r] :
( mem(V3r,A_27a)
=> ! [V4s] :
( mem(V4s,A_27a)
=> ( p(ap(ap(V0R,V3r),V4s))
<=> ( p(ap(ap(V0R,V3r),V3r))
& p(ap(ap(V0R,V4s),V4s))
& ap(V1abs,V3r) = ap(V1abs,V4s) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Equotient_2EFUN__QUOTIENT,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))
=> p(ap(ap(ap(c_2Equotient_2EQUOTIENT(arr(A_27a,A_27b),arr(A_27c,A_27d)),ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,A_27b),V0R1),V3R2)),ap(ap(c_2Equotient_2E_2D_2D_3E(A_27c,A_27b,A_27a,A_27d),V2rep1),V4abs2)),ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,A_27d,A_27c,A_27b),V1abs1),V5rep2))) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2Equotient_2Erespects__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2Equotient_2Erespects(A_27a,A_27b) = c_2Ecombin_2EW(A_27a,A_27b) ) ) ).
fof(conj_thm_2Equotient_2EFUN__REL__EQ__REL,conjecture,
! [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))
=> ! [V6f] :
( mem(V6f,arr(A_27a,A_27b))
=> ! [V7g] :
( mem(V7g,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,A_27b),V0R1),V3R2),V6f),V7g))
<=> ( p(ap(ap(c_2Equotient_2Erespects(arr(A_27a,A_27b),bool),ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,A_27b),V0R1),V3R2)),V6f))
& p(ap(ap(c_2Equotient_2Erespects(arr(A_27a,A_27b),bool),ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,A_27b),V0R1),V3R2)),V7g))
& ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27c,A_27b,A_27a,A_27d),V2rep1),V4abs2),V6f) = ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27c,A_27b,A_27a,A_27d),V2rep1),V4abs2),V7g) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------