TPTP Problem File: ITP006^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP006^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT.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_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT.p [Gau19]
% : HL402501^2.p [TPAP]
% Status : Theorem
% Rating : 0.60 v8.2.0, 0.62 v8.1.0, 0.64 v7.5.0
% Syntax : Number of formulae : 75 ( 4 unt; 24 typ; 0 def)
% Number of atoms : 450 ( 10 equ; 0 cnn)
% Maximal formula atoms : 75 ( 8 avg)
% Number of connectives : 833 ( 40 ~; 22 |; 30 &; 581 @)
% ( 46 <=>; 114 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 9 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 28 ( 28 >; 0 *; 0 +; 0 <<)
% Number of symbols : 31 ( 28 usr; 15 con; 0-2 aty)
% Number of variables : 113 ( 0 ^; 108 !; 5 ?; 113 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
thf(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
thf(mem_c_2Ebool_2ET,axiom,
mem @ c_2Ebool_2ET @ bool ).
thf(ax_true_p,axiom,
p @ c_2Ebool_2ET ).
thf(tp_c_2EquantHeuristics_2EGUESS__FORALL__GAP,type,
c_2EquantHeuristics_2EGUESS__FORALL__GAP: del > del > $i ).
thf(mem_c_2EquantHeuristics_2EGUESS__FORALL__GAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2EquantHeuristics_2EGUESS__EXISTS__GAP,type,
c_2EquantHeuristics_2EGUESS__EXISTS__GAP: del > del > $i ).
thf(mem_c_2EquantHeuristics_2EGUESS__EXISTS__GAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2EquantHeuristics_2EGUESS__FORALL__POINT,type,
c_2EquantHeuristics_2EGUESS__FORALL__POINT: del > del > $i ).
thf(mem_c_2EquantHeuristics_2EGUESS__FORALL__POINT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2EquantHeuristics_2EGUESS__EXISTS__POINT,type,
c_2EquantHeuristics_2EGUESS__EXISTS__POINT: del > del > $i ).
thf(mem_c_2EquantHeuristics_2EGUESS__EXISTS__POINT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2EquantHeuristics_2EGUESS__FORALL,type,
c_2EquantHeuristics_2EGUESS__FORALL: del > del > $i ).
thf(mem_c_2EquantHeuristics_2EGUESS__FORALL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
thf(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_3F @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_ex_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A ) @ Q ) )
<=> ? [X: $i] :
( ( mem @ X @ A )
& ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(tp_c_2EquantHeuristics_2EGUESS__EXISTS,type,
c_2EquantHeuristics_2EGUESS__EXISTS: del > del > $i ).
thf(mem_c_2EquantHeuristics_2EGUESS__EXISTS,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
thf(mem_c_2Ebool_2EF,axiom,
mem @ c_2Ebool_2EF @ bool ).
thf(ax_false_p,axiom,
~ ( p @ c_2Ebool_2EF ) ).
thf(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
thf(mem_c_2Ebool_2E_5C_2F,axiom,
mem @ c_2Ebool_2E_5C_2F @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_or_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ Q ) @ R ) )
<=> ( ( p @ Q )
| ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
thf(mem_c_2Ebool_2E_2F_5C,axiom,
mem @ c_2Ebool_2E_2F_5C @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_and_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ Q ) @ R ) )
<=> ( ( p @ Q )
& ( p @ R ) ) ) ) ) ).
thf(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
thf(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : ( mem @ ( c_2Emin_2E_3D @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ).
thf(ax_eq_p,axiom,
! [A: del,X: $i] :
( ( mem @ X @ A )
=> ! [Y: $i] :
( ( mem @ Y @ A )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A ) @ X ) @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
thf(mem_c_2Ebool_2E_7E,axiom,
mem @ c_2Ebool_2E_7E @ ( arr @ bool @ bool ) ).
thf(ax_neg_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2E_7E @ Q ) )
<=> ~ ( p @ Q ) ) ) ).
thf(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
thf(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem @ c_2Emin_2E_3D_3D_3E @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_imp_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ Q ) @ R ) )
<=> ( ( p @ Q )
=> ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
thf(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_all_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A ) @ Q ) )
<=> ! [X: $i] :
( ( mem @ X @ A )
=> ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
thf(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
thf(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( V0x = V1y )
<=> ( V1y = V0x ) ) ) ) ).
thf(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( $true
<=> ( p @ V0t ) )
<=> ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $true )
<=> ( p @ V0t ) )
& ( ( $false
<=> ( p @ V0t ) )
<=> ~ ( p @ V0t ) )
& ( ( ( p @ V0t )
<=> $false )
<=> ~ ( p @ V0t ) ) ) ) ).
thf(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: $i] :
( ( mem @ V0t1 @ bool )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ bool )
=> ! [V2t3: $i] :
( ( mem @ V2t3 @ bool )
=> ( ( ( p @ V0t1 )
=> ( ( p @ V1t2 )
=> ( p @ V2t3 ) ) )
<=> ( ( ( p @ V0t1 )
& ( p @ V1t2 ) )
=> ( p @ V2t3 ) ) ) ) ) ) ).
thf(conj_thm_2Ebool_2EIMP__CONG,axiom,
! [V0x: $i] :
( ( mem @ V0x @ bool )
=> ! [V1x_27: $i] :
( ( mem @ V1x_27 @ bool )
=> ! [V2y: $i] :
( ( mem @ V2y @ bool )
=> ! [V3y_27: $i] :
( ( mem @ V3y_27 @ bool )
=> ( ( ( ( p @ V0x )
<=> ( p @ V1x_27 ) )
& ( ( p @ V1x_27 )
=> ( ( p @ V2y )
<=> ( p @ V3y_27 ) ) ) )
=> ( ( ( p @ V0x )
=> ( p @ V2y ) )
<=> ( ( p @ V1x_27 )
=> ( p @ V3y_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EquantHeuristics_2EGUESS__REWRITES,axiom,
! [A_27a: del,A_27b: del,V0i: $i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ( ( p @ ( ap @ V1P @ V2v ) )
=> ? [V3fv: $i] :
( ( mem @ V3fv @ A_27a )
& ( p @ ( ap @ V1P @ ( ap @ V0i @ V3fv ) ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V4v: $i] :
( ( mem @ V4v @ A_27b )
=> ( ~ ( p @ ( ap @ V1P @ V4v ) )
=> ? [V5fv: $i] :
( ( mem @ V5fv @ A_27a )
& ~ ( p @ ( ap @ V1P @ ( ap @ V0i @ V5fv ) ) ) ) ) ) )
& ! [V6i: $i] :
( ( mem @ V6i @ ( arr @ A_27a @ A_27b ) )
=> ! [V7P: $i] :
( ( mem @ V7P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V6i ) @ V7P ) )
<=> ! [V8fv: $i] :
( ( mem @ V8fv @ A_27a )
=> ( p @ ( ap @ V7P @ ( ap @ V6i @ V8fv ) ) ) ) ) ) )
& ! [V9i: $i] :
( ( mem @ V9i @ ( arr @ A_27a @ A_27b ) )
=> ! [V10P: $i] :
( ( mem @ V10P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V9i ) @ V10P ) )
<=> ! [V11fv: $i] :
( ( mem @ V11fv @ A_27a )
=> ~ ( p @ ( ap @ V10P @ ( ap @ V9i @ V11fv ) ) ) ) ) ) )
& ! [V12i: $i] :
( ( mem @ V12i @ ( arr @ A_27a @ A_27b ) )
=> ! [V13P: $i] :
( ( mem @ V13P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V12i ) @ V13P ) )
<=> ! [V14v: $i] :
( ( mem @ V14v @ A_27b )
=> ( ( p @ ( ap @ V13P @ V14v ) )
=> ? [V15fv: $i] :
( ( mem @ V15fv @ A_27a )
& ( V14v
= ( ap @ V12i @ V15fv ) ) ) ) ) ) ) )
& ! [V16i: $i] :
( ( mem @ V16i @ ( arr @ A_27a @ A_27b ) )
=> ! [V17P: $i] :
( ( mem @ V17P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V16i ) @ V17P ) )
<=> ! [V18v: $i] :
( ( mem @ V18v @ A_27b )
=> ( ~ ( p @ ( ap @ V17P @ V18v ) )
=> ? [V19fv: $i] :
( ( mem @ V19fv @ A_27a )
& ( V18v
= ( ap @ V16i @ V19fv ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( p @ V0A )
=> ( ~ ( p @ V0A )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( ( p @ V0A )
=> $false )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ! [V1B: $i] :
( ( mem @ V1B @ bool )
=> ( ( ~ ( ~ ( p @ V0A )
| ( p @ V1B ) )
=> $false )
<=> ( ( p @ V0A )
=> ( ~ ( p @ V1B )
=> $false ) ) ) ) ) ).
thf(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A: $i] :
( ( mem @ V0A @ bool )
=> ( ( ~ ( p @ V0A )
=> $false )
=> ( ( ( p @ V0A )
=> $false )
=> $false ) ) ) ).
thf(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( 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 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
| ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ~ ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ! [V2r: $i] :
( ( mem @ V2r @ bool )
=> ( ( ( p @ V0p )
<=> ( ( p @ V1q )
=> ( p @ V2r ) ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ( p @ V0p )
| ~ ( p @ V2r ) )
& ( ~ ( p @ V1q )
| ( p @ V2r )
| ~ ( p @ V0p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ( ( p @ V0p )
<=> ~ ( p @ V1q ) )
<=> ( ( ( p @ V0p )
| ( p @ V1q ) )
& ( ~ ( p @ V1q )
| ~ ( p @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__ni1,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
=> ( p @ V1q ) )
=> ( p @ V0p ) ) ) ) ).
thf(conj_thm_2Esat_2Epth__ni2,axiom,
! [V0p: $i] :
( ( mem @ V0p @ bool )
=> ! [V1q: $i] :
( ( mem @ V1q @ bool )
=> ( ~ ( ( p @ V0p )
=> ( p @ V1q ) )
=> ~ ( p @ V1q ) ) ) ) ).
thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT,conjecture,
! [A_27a: del,A_27b: del,V0i: $i] :
( ( mem @ V0i @ ( arr @ A_27b @ A_27a ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V2Q @ V3x ) )
=> ( p @ ( ap @ V1P @ V3x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27a ) @ V0i ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27a ) @ V0i ) @ V2Q ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------