TPTP Problem File: ITP020+2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP020+2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Eutil__prob_2ENUM__2D__BIJ__INV.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_2Eutil__prob_2ENUM__2D__BIJ__INV.p [Gau19]
% : HL409501+2.p [TPAP]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.75 v8.1.0, 0.72 v7.5.0
% Syntax : Number of formulae : 63 ( 12 unt; 0 def)
% Number of atoms : 339 ( 8 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 317 ( 41 ~; 36 |; 45 &)
% ( 52 <=>; 143 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 20 ( 20 usr; 9 con; 0-2 aty)
% Number of variables : 136 ( 120 !; 16 ?)
% 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_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
fof(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
fof(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
fof(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
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_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_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_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_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(mem_c_2Epred__set_2EUNIV,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Epred__set_2EUNIV(A_27a),arr(A_27a,bool)) ) ).
fof(ne_ty_2Epair_2Eprod,axiom,
! [A0] :
( ne(A0)
=> ! [A1] :
( ne(A1)
=> ne(ty_2Epair_2Eprod(A0,A1)) ) ) ).
fof(mem_c_2Epred__set_2ECROSS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epred__set_2ECROSS(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_2Epred__set_2EBIJ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Epred__set_2EBIJ(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27a,bool),arr(arr(A_27b,bool),bool)))) ) ) ).
fof(ne_ty_2Enum_2Enum,axiom,
ne(ty_2Enum_2Enum) ).
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(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_2EIMP__F,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
=> $false )
=> ~ p(V0t) ) ) ).
fof(conj_thm_2Ebool_2EF__IMP,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ p(V0t)
=> ( p(V0t)
=> $false ) ) ) ).
fof(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( ( $true
=> p(V0t) )
<=> p(V0t) )
& ( ( p(V0t)
=> $true )
<=> $true )
& ( ( $false
=> p(V0t) )
<=> $true )
& ( ( p(V0t)
=> p(V0t) )
<=> $true )
& ( ( p(V0t)
=> $false )
<=> ~ 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_2EEQ__SYM__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0x] :
( mem(V0x,A_27a)
=> ! [V1y] :
( mem(V1y,A_27a)
=> ( V0x = V1y
<=> V1y = V0x ) ) ) ) ).
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_2ENOT__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ( ~ ? [V1x] :
( mem(V1x,A_27a)
& p(ap(V0P,V1x)) )
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ~ p(ap(V0P,V2x)) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__AND__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V0P,V2x)) )
& p(V1Q) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(ap(V0P,V3x))
& p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__AND__FORALL__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( p(V0P)
& ! [V2x] :
( mem(V2x,A_27a)
=> p(ap(V1Q,V2x)) ) )
<=> ! [V3x] :
( mem(V3x,A_27a)
=> ( p(V0P)
& p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__OR__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x)) )
| p(V1Q) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& ( p(ap(V0P,V3x))
| p(V1Q) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__OR__EXISTS__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ( p(V0P)
| ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V1Q,V2x)) ) )
<=> ? [V3x] :
( mem(V3x,A_27a)
& ( p(V0P)
| p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__EXISTS__AND__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,bool)
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(ap(V0P,V2x))
& p(V1Q) )
<=> ( ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V0P,V3x)) )
& p(V1Q) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__EXISTS__AND__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ? [V2x] :
( mem(V2x,A_27a)
& p(V0P)
& p(ap(V1Q,V2x)) )
<=> ( p(V0P)
& ? [V3x] :
( mem(V3x,A_27a)
& p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ELEFT__FORALL__OR__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0Q] :
( mem(V0Q,bool)
=> ! [V1P] :
( mem(V1P,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V1P,V2x))
| p(V0Q) ) )
<=> ( ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V1P,V3x)) )
| p(V0Q) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2ERIGHT__FORALL__OR__THM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,bool)
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2x] :
( mem(V2x,A_27a)
=> ( p(V0P)
| p(ap(V1Q,V2x)) ) )
<=> ( p(V0P)
| ! [V3x] :
( mem(V3x,A_27a)
=> p(ap(V1Q,V3x)) ) ) ) ) ) ) ).
fof(conj_thm_2Ebool_2EDISJ__SYM,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( p(V0A)
| p(V1B) )
<=> ( p(V1B)
| p(V0A) ) ) ) ) ).
fof(conj_thm_2Ebool_2ESKOLEM__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)) ) )
<=> ? [V3f] :
( mem(V3f,arr(A_27a,A_27b))
& ! [V4x] :
( mem(V4x,A_27a)
=> p(ap(ap(V0P,V4x),ap(V3f,V4x))) ) ) ) ) ) ) ).
fof(conj_thm_2Epred__set_2EBIJ__SYM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( mem(V1t,arr(A_27b,bool))
=> ( ? [V2f] :
( mem(V2f,arr(A_27a,A_27b))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27a,A_27b),V2f),V0s),V1t)) )
<=> ? [V3g] :
( mem(V3g,arr(A_27b,A_27a))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(A_27b,A_27a),V3g),V1t),V0s)) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ ~ p(V0t)
<=> p(V0t) ) ) ).
fof(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A] :
( mem(V0A,bool)
=> ( p(V0A)
=> ( ~ p(V0A)
=> $false ) ) ) ).
fof(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( p(V0A)
| p(V1B) )
=> $false )
<=> ( ( p(V0A)
=> $false )
=> ( ~ p(V1B)
=> $false ) ) ) ) ) ).
fof(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A] :
( mem(V0A,bool)
=> ! [V1B] :
( mem(V1B,bool)
=> ( ( ~ ( ~ p(V0A)
| p(V1B) )
=> $false )
<=> ( p(V0A)
=> ( ~ p(V1B)
=> $false ) ) ) ) ) ).
fof(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A] :
( mem(V0A,bool)
=> ( ( ~ p(V0A)
=> $false )
=> ( ( p(V0A)
=> $false )
=> $false ) ) ) ).
fof(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
<=> p(V2r) ) )
<=> ( ( p(V0p)
| p(V1q)
| p(V2r) )
& ( p(V0p)
| ~ p(V2r)
| ~ p(V1q) )
& ( p(V1q)
| ~ p(V2r)
| ~ p(V0p) )
& ( p(V2r)
| ~ p(V1q)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__conj,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
& p(V2r) ) )
<=> ( ( p(V0p)
| ~ p(V1q)
| ~ p(V2r) )
& ( p(V1q)
| ~ p(V0p) )
& ( p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
| p(V2r) ) )
<=> ( ( p(V0p)
| ~ p(V1q) )
& ( p(V0p)
| ~ p(V2r) )
& ( p(V1q)
| p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ! [V2r] :
( mem(V2r,bool)
=> ( ( p(V0p)
<=> ( p(V1q)
=> p(V2r) ) )
<=> ( ( p(V0p)
| p(V1q) )
& ( p(V0p)
| ~ p(V2r) )
& ( ~ p(V1q)
| p(V2r)
| ~ p(V0p) ) ) ) ) ) ) ).
fof(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p] :
( mem(V0p,bool)
=> ! [V1q] :
( mem(V1q,bool)
=> ( ( p(V0p)
<=> ~ p(V1q) )
<=> ( ( p(V0p)
| p(V1q) )
& ( ~ p(V1q)
| ~ p(V0p) ) ) ) ) ) ).
fof(conj_thm_2Eutil__prob_2ENUM__2D__BIJ,axiom,
? [V0f] :
( mem(V0f,arr(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum),ty_2Enum_2Enum),V0f),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ).
fof(conj_thm_2Eutil__prob_2ENUM__2D__BIJ__INV,conjecture,
? [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)))
& p(ap(ap(ap(c_2Epred__set_2EBIJ(ty_2Enum_2Enum,ty_2Epair_2Eprod(ty_2Enum_2Enum,ty_2Enum_2Enum)),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),ap(ap(c_2Epred__set_2ECROSS(ty_2Enum_2Enum,ty_2Enum_2Enum),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ).
%------------------------------------------------------------------------------