## TPTP Problem File: ITP010^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : ITP010^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : HOL4 set theory export of thm_2Ecardinal_2ECARD__NOT__LE.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_2Ecardinal_2ECARD__NOT__LE.p [Gau19]
%          : HL404501^2.p [TPAP]

% Status   : Theorem
% Rating   : 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax   : Number of formulae    :   58 (   4 unt;  18 typ;   0 def)
%            Number of atoms       :  264 (   7 equ;   0 cnn)
%            Maximal formula atoms :   21 (   6 avg)
%            Number of connectives :  459 (  29   ~;  14   |;  14   &; 305   @)
%                                         (  33 <=>;  64  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   8 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   17 (  17   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   25 (  22 usr;  15 con; 0-2 aty)
%            Number of variables   :   65 (   0   ^;  65   !;   0   ?;  65   :)
% SPC      : TH0_THM_EQU_NAR

% 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_2Ecardinal_2Ecardleq,type,
c_2Ecardinal_2Ecardleq: del > del > \$i ).

thf(mem_c_2Ecardinal_2Ecardleq,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( 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_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_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_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,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_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_2EFALSITY,axiom,
! [V0t: \$i] :
( ( mem @ V0t @ bool )
=> ( \$false
=> ( p @ V0t ) ) ) ).

thf(conj_thm_2Ebool_2EFORALL__SIMP,axiom,
! [A_27a: del,V0t: \$i] :
( ( mem @ V0t @ bool )
=> ( ! [V1x: \$i] :
( ( mem @ V1x @ A_27a )
=> ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).

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_2EREFL__CLAUSE,axiom,
! [A_27a: del,V0x: \$i] :
( ( mem @ V0x @ A_27a )
=> ( ( V0x = V0x )
<=> \$true ) ) ).

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_2Ecardinal_2ECARD__LE__TOTAL,axiom,
! [A_27a: del,A_27b: del,V0s: \$i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: \$i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
| ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) ).

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__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_2Ecardinal_2ECARD__NOT__LE,conjecture,
! [A_27a: del,A_27b: del,V0s: \$i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: \$i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
<=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) ).

%------------------------------------------------------------------------------
```