ITP001 Axioms: ITP004^5.ax
%------------------------------------------------------------------------------
% File : ITP004^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : ConseqConv^2.ax [Gau20]
% : HL4004^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 43 ( 1 unt; 1 typ; 0 def)
% Number of atoms : 416 ( 4 equ; 0 cnn)
% Maximal formula atoms : 28 ( 9 avg)
% Number of connectives : 663 ( 14 ~; 9 |; 23 &; 431 @)
% ( 27 <=>; 159 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg; 431 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 14 ( 11 usr; 13 con; 0-2 aty)
% Number of variables : 93 ( 2 ^ 89 !; 2 ?; 93 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2EConseqConv_2EASM__MARKER,type,
c_2EConseqConv_2EASM__MARKER: $i ).
thf(mem_c_2EConseqConv_2EASM__MARKER,axiom,
mem @ c_2EConseqConv_2EASM__MARKER @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(conj_thm_2EConseqConv_2Eforall__eq__thm,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2s ) )
<=> ( p @ ( ap @ V1Q @ V2s ) ) ) )
=> ( ! [V3s: $i] :
( ( mem @ V3s @ A_27a )
=> ( p @ ( ap @ V0P @ V3s ) ) )
<=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( p @ ( ap @ V1Q @ V4s ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2Eexists__eq__thm,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2s: $i] :
( ( mem @ V2s @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2s ) )
<=> ( p @ ( ap @ V1Q @ V2s ) ) ) )
=> ( ? [V3s: $i] :
( ( mem @ V3s @ A_27a )
& ( p @ ( ap @ V0P @ V3s ) ) )
<=> ? [V4s: $i] :
( ( mem @ V4s @ A_27a )
& ( p @ ( ap @ V1Q @ V4s ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2Etrue__imp,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( p @ V0t )
=> $true ) ) ).
thf(conj_thm_2EConseqConv_2Efalse__imp,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( $false
=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2ENOT__CLAUSES__X,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ~ ~ ( p @ V0t )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2ENOT__CLAUSES__T,axiom,
( ~ $true
<=> $false ) ).
thf(conj_thm_2EConseqConv_2ENOT__CLAUSES__F,axiom,
( ~ $false
<=> $true ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__conj__strengthen,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 @ V2y )
=> ( ( p @ V1x_27 )
=> ( p @ V0x ) ) )
& ( ( p @ V1x_27 )
=> ( ( p @ V3y_27 )
=> ( p @ V2y ) ) ) )
=> ( ( ( p @ V1x_27 )
& ( p @ V3y_27 ) )
=> ( ( p @ V0x )
& ( p @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__conj__weaken,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 @ V2y )
=> ( ( 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_2EConseqConv_2EAND__CLAUSES__TX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( $true
& ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2EAND__CLAUSES__XT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
& $true )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2EAND__CLAUSES__FX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( $false
& ( p @ V0t ) )
<=> $false ) ) ).
thf(conj_thm_2EConseqConv_2EAND__CLAUSES__XF,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
& $false )
<=> $false ) ) ).
thf(conj_thm_2EConseqConv_2EAND__CLAUSES__XX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
& ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__disj__strengthen,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 @ V2y )
=> ( ( p @ V1x_27 )
=> ( p @ V0x ) ) )
& ( ~ ( p @ V1x_27 )
=> ( ( p @ V3y_27 )
=> ( p @ V2y ) ) ) )
=> ( ( ( p @ V1x_27 )
| ( p @ V3y_27 ) )
=> ( ( p @ V0x )
| ( p @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__disj__weaken,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 @ V2y )
=> ( ( 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_2EConseqConv_2EOR__CLAUSES__TX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( $true
| ( p @ V0t ) )
<=> $true ) ) ).
thf(conj_thm_2EConseqConv_2EOR__CLAUSES__XT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
| $true )
<=> $true ) ) ).
thf(conj_thm_2EConseqConv_2EOR__CLAUSES__FX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( $false
| ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2EOR__CLAUSES__XF,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
| $false )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2EOR__CLAUSES__XX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
| ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__imp__strengthen,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 @ V3y_27 )
=> ( p @ V2y ) ) )
& ( ~ ( p @ V3y_27 )
=> ( ( p @ V0x )
=> ( p @ V1x_27 ) ) ) )
=> ( ( ( p @ V1x_27 )
=> ( p @ V3y_27 ) )
=> ( ( p @ V0x )
=> ( p @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__imp__weaken,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 @ V2y )
=> ( p @ V3y_27 ) ) )
& ( ~ ( p @ V3y_27 )
=> ( ( p @ V1x_27 )
=> ( p @ V0x ) ) ) )
=> ( ( ( p @ V0x )
=> ( p @ V2y ) )
=> ( ( p @ V1x_27 )
=> ( p @ V3y_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__simple__imp__strengthen,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 @ V3y_27 )
=> ( p @ V2y ) ) ) )
=> ( ( ( p @ V1x_27 )
=> ( p @ V3y_27 ) )
=> ( ( p @ V0x )
=> ( p @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__simple__imp__weaken,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 @ V1x_27 )
=> ( p @ V0x ) )
& ( ( p @ V1x_27 )
=> ( ( p @ V2y )
=> ( p @ V3y_27 ) ) ) )
=> ( ( ( p @ V0x )
=> ( p @ V2y ) )
=> ( ( p @ V1x_27 )
=> ( p @ V3y_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CLAUSES__TX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( $true
=> ( p @ V0t ) )
<=> ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CLAUSES__XT,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
=> $true )
<=> $true ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CLAUSES__FX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( $false
=> ( p @ V0t ) )
<=> $true ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CLAUSES__XX,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
=> ( p @ V0t ) )
<=> $true ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CLAUSES__XF,axiom,
! [V0t: $i] :
( ( mem @ V0t @ bool )
=> ( ( ( p @ V0t )
=> $false )
<=> ~ ( p @ V0t ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__cond__simple,axiom,
! [V0c: $i] :
( ( mem @ V0c @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ! [V2x_27: $i] :
( ( mem @ V2x_27 @ bool )
=> ! [V3y: $i] :
( ( mem @ V3y @ bool )
=> ! [V4y_27: $i] :
( ( mem @ V4y_27 @ bool )
=> ( ( ( ( p @ V2x_27 )
=> ( p @ V1x ) )
& ( ( p @ V4y_27 )
=> ( p @ V3y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0c ) @ V2x_27 ) @ V4y_27 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0c ) @ V1x ) @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2EIMP__CONG__cond,axiom,
! [V0c: $i] :
( ( mem @ V0c @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ! [V2x_27: $i] :
( ( mem @ V2x_27 @ bool )
=> ! [V3y: $i] :
( ( mem @ V3y @ bool )
=> ! [V4y_27: $i] :
( ( mem @ V4y_27 @ bool )
=> ( ( ( ( p @ V0c )
=> ( ( p @ V2x_27 )
=> ( p @ V1x ) ) )
& ( ~ ( p @ V0c )
=> ( ( p @ V4y_27 )
=> ( p @ V3y ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0c ) @ V2x_27 ) @ V4y_27 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0c ) @ V1x ) @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2ECOND__CLAUSES__CT,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ A_27a )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ c_2Ebool_2ET ) @ V0t1 ) @ V1t2 )
= V0t1 ) ) ) ).
thf(conj_thm_2EConseqConv_2ECOND__CLAUSES__CF,axiom,
! [A_27a: del,V0t1: $i] :
( ( mem @ V0t1 @ A_27a )
=> ! [V1t2: $i] :
( ( mem @ V1t2 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ c_2Ebool_2EF ) @ V0t1 ) @ V1t2 )
= V1t2 ) ) ) ).
thf(conj_thm_2EConseqConv_2ECOND__CLAUSES__ID,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ bool )
=> ! [V1t: $i] :
( ( mem @ V1t @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V0b ) @ V1t ) @ V1t )
= V1t ) ) ) ).
thf(conj_thm_2EConseqConv_2ECOND__CLAUSES__TT,axiom,
! [V0c: $i] :
( ( mem @ V0c @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0c ) @ c_2Ebool_2ET ) @ V1x ) )
<=> ( ~ ( p @ V0c )
=> ( p @ V1x ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2ECOND__CLAUSES__FT,axiom,
! [V0c: $i] :
( ( mem @ V0c @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0c ) @ V1x ) @ c_2Ebool_2ET ) )
<=> ( ( p @ V0c )
=> ( p @ V1x ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2ECOND__CLAUSES__TF,axiom,
! [V0c: $i] :
( ( mem @ V0c @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0c ) @ c_2Ebool_2EF ) @ V1x ) )
<=> ( ~ ( p @ V0c )
& ( p @ V1x ) ) ) ) ) ).
thf(conj_thm_2EConseqConv_2ECOND__CLAUSES__FF,axiom,
! [V0c: $i] :
( ( mem @ V0c @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V0c ) @ V1x ) @ c_2Ebool_2EF ) )
<=> ( ( p @ V0c )
& ( p @ V1x ) ) ) ) ) ).
thf(ax_thm_2EConseqConv_2EASM__MARKER__DEF,axiom,
( c_2EConseqConv_2EASM__MARKER
= ( lam @ bool
@ ^ [V0y: $i] :
( lam @ bool
@ ^ [V1x: $i] : V1x ) ) ) ).
thf(conj_thm_2EConseqConv_2EASM__MARKER__THM,axiom,
! [V0y: $i] :
( ( mem @ V0y @ bool )
=> ! [V1x: $i] :
( ( mem @ V1x @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2EConseqConv_2EASM__MARKER @ V0y ) @ V1x ) )
<=> ( p @ V1x ) ) ) ) ).
%------------------------------------------------------------------------------