ITP001 Axioms: ITP004^7.ax
%------------------------------------------------------------------------------
% File : ITP004^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy 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 : ConseqConv.ax [Gau19]
% : HL4004^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 61 ( 25 unt; 13 typ; 0 def)
% Number of atoms : 63 ( 8 equ; 15 cnn)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 239 ( 15 ~; 10 |; 22 &; 85 @)
% ( 31 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg; 85 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 26 ( 26 >; 0 *; 0 +; 0 <<)
% Number of symbols : 14 ( 12 usr; 3 con; 0-4 aty)
% Number of variables : 113 ( 2 ^ 104 !; 3 ?; 113 :)
% ( 4 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2EConseqConv_2EASM__MARKER,type,
c_2EConseqConv_2EASM__MARKER: $o > $o > $o ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2EConseqConv_2EASM__MARKER__DEF,axiom,
( c_2EConseqConv_2EASM__MARKER
= ( ^ [V0y: $o,V1x: $o] : V1x ) ) ).
thf(thm_2EConseqConv_2Eforall__eq__thm,axiom,
! [A_27a: $tType,V0Q: A_27a > $o,V1P: A_27a > $o] :
( ! [V2s: A_27a] :
( ( V1P @ V2s )
= ( V0Q @ V2s ) )
=> ( ! [V3s: A_27a] : ( V1P @ V3s )
<=> ! [V4s: A_27a] : ( V0Q @ V4s ) ) ) ).
thf(thm_2EConseqConv_2Eexists__eq__thm,axiom,
! [A_27a: $tType,V0Q: A_27a > $o,V1P: A_27a > $o] :
( ! [V2s: A_27a] :
( ( V1P @ V2s )
= ( V0Q @ V2s ) )
=> ( ? [V3s: A_27a] : ( V1P @ V3s )
<=> ? [V4s: A_27a] : ( V0Q @ V4s ) ) ) ).
thf(thm_2EConseqConv_2Etrue__imp,axiom,
! [V0t: $o] :
( V0t
=> c_2Ebool_2ET ) ).
thf(thm_2EConseqConv_2Efalse__imp,axiom,
! [V0t: $o] :
( c_2Ebool_2EF
=> V0t ) ).
thf(thm_2EConseqConv_2ENOT__CLAUSES__X,axiom,
! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t ) ).
thf(thm_2EConseqConv_2ENOT__CLAUSES__T,axiom,
( ( (~) @ c_2Ebool_2ET )
<=> c_2Ebool_2EF ) ).
thf(thm_2EConseqConv_2ENOT__CLAUSES__F,axiom,
( ( (~) @ c_2Ebool_2EF )
<=> c_2Ebool_2ET ) ).
thf(thm_2EConseqConv_2EIMP__CONG__conj__strengthen,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V2y
=> ( V1x_27
=> V0x ) )
& ( V1x_27
=> ( V3y_27
=> V2y ) ) )
=> ( ( V1x_27
& V3y_27 )
=> ( V0x
& V2y ) ) ) ).
thf(thm_2EConseqConv_2EIMP__CONG__conj__weaken,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V2y
=> ( V0x
=> V1x_27 ) )
& ( V1x_27
=> ( V2y
=> V3y_27 ) ) )
=> ( ( V0x
& V2y )
=> ( V1x_27
& V3y_27 ) ) ) ).
thf(thm_2EConseqConv_2EAND__CLAUSES__TX,axiom,
! [V0t: $o] :
( ( c_2Ebool_2ET
& V0t )
<=> V0t ) ).
thf(thm_2EConseqConv_2EAND__CLAUSES__XT,axiom,
! [V0t: $o] :
( ( V0t
& c_2Ebool_2ET )
<=> V0t ) ).
thf(thm_2EConseqConv_2EAND__CLAUSES__FX,axiom,
! [V0t: $o] :
( ( c_2Ebool_2EF
& V0t )
<=> c_2Ebool_2EF ) ).
thf(thm_2EConseqConv_2EAND__CLAUSES__XF,axiom,
! [V0t: $o] :
( ( V0t
& c_2Ebool_2EF )
<=> c_2Ebool_2EF ) ).
thf(thm_2EConseqConv_2EAND__CLAUSES__XX,axiom,
! [V0t: $o] :
( ( V0t
& V0t )
<=> V0t ) ).
thf(thm_2EConseqConv_2EIMP__CONG__disj__strengthen,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( ( (~) @ V2y )
=> ( V1x_27
=> V0x ) )
& ( ( (~) @ V1x_27 )
=> ( V3y_27
=> V2y ) ) )
=> ( ( V1x_27
| V3y_27 )
=> ( V0x
| V2y ) ) ) ).
thf(thm_2EConseqConv_2EIMP__CONG__disj__weaken,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( ( (~) @ V2y )
=> ( V0x
=> V1x_27 ) )
& ( ( (~) @ V1x_27 )
=> ( V2y
=> V3y_27 ) ) )
=> ( ( V0x
| V2y )
=> ( V1x_27
| V3y_27 ) ) ) ).
thf(thm_2EConseqConv_2EOR__CLAUSES__TX,axiom,
! [V0t: $o] :
( ( c_2Ebool_2ET
| V0t )
<=> c_2Ebool_2ET ) ).
thf(thm_2EConseqConv_2EOR__CLAUSES__XT,axiom,
! [V0t: $o] :
( ( V0t
| c_2Ebool_2ET )
<=> c_2Ebool_2ET ) ).
thf(thm_2EConseqConv_2EOR__CLAUSES__FX,axiom,
! [V0t: $o] :
( ( c_2Ebool_2EF
| V0t )
<=> V0t ) ).
thf(thm_2EConseqConv_2EOR__CLAUSES__XF,axiom,
! [V0t: $o] :
( ( V0t
| c_2Ebool_2EF )
<=> V0t ) ).
thf(thm_2EConseqConv_2EOR__CLAUSES__XX,axiom,
! [V0t: $o] :
( ( V0t
| V0t )
<=> V0t ) ).
thf(thm_2EConseqConv_2EIMP__CONG__imp__strengthen,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V0x
=> ( V3y_27
=> V2y ) )
& ( ( (~) @ V3y_27 )
=> ( V0x
=> V1x_27 ) ) )
=> ( ( V1x_27
=> V3y_27 )
=> ( V0x
=> V2y ) ) ) ).
thf(thm_2EConseqConv_2EIMP__CONG__imp__weaken,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V0x
=> ( V2y
=> V3y_27 ) )
& ( ( (~) @ V3y_27 )
=> ( V1x_27
=> V0x ) ) )
=> ( ( V0x
=> V2y )
=> ( V1x_27
=> V3y_27 ) ) ) ).
thf(thm_2EConseqConv_2EIMP__CONG__simple__imp__strengthen,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V0x
=> V1x_27 )
& ( V1x_27
=> ( V3y_27
=> V2y ) ) )
=> ( ( V1x_27
=> V3y_27 )
=> ( V0x
=> V2y ) ) ) ).
thf(thm_2EConseqConv_2EIMP__CONG__simple__imp__weaken,axiom,
! [V0x: $o,V1x_27: $o,V2y: $o,V3y_27: $o] :
( ( ( V1x_27
=> V0x )
& ( V1x_27
=> ( V2y
=> V3y_27 ) ) )
=> ( ( V0x
=> V2y )
=> ( V1x_27
=> V3y_27 ) ) ) ).
thf(thm_2EConseqConv_2EIMP__CLAUSES__TX,axiom,
! [V0t: $o] :
( ( c_2Ebool_2ET
=> V0t )
<=> V0t ) ).
thf(thm_2EConseqConv_2EIMP__CLAUSES__XT,axiom,
! [V0t: $o] :
( ( V0t
=> c_2Ebool_2ET )
<=> c_2Ebool_2ET ) ).
thf(thm_2EConseqConv_2EIMP__CLAUSES__FX,axiom,
! [V0t: $o] :
( ( c_2Ebool_2EF
=> V0t )
<=> c_2Ebool_2ET ) ).
thf(thm_2EConseqConv_2EIMP__CLAUSES__XX,axiom,
! [V0t: $o] :
( ( V0t
=> V0t )
<=> c_2Ebool_2ET ) ).
thf(thm_2EConseqConv_2EIMP__CLAUSES__XF,axiom,
! [V0t: $o] :
( ( V0t
=> c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ).
thf(thm_2EConseqConv_2EIMP__CONG__cond__simple,axiom,
! [V0c: $o,V1x: $o,V2x_27: $o,V3y: $o,V4y_27: $o] :
( ( ( V2x_27
=> V1x )
& ( V4y_27
=> V3y ) )
=> ( ( c_2Ebool_2ECOND @ $o @ V0c @ V2x_27 @ V4y_27 )
=> ( c_2Ebool_2ECOND @ $o @ V0c @ V1x @ V3y ) ) ) ).
thf(thm_2EConseqConv_2EIMP__CONG__cond,axiom,
! [V0c: $o,V1x: $o,V2x_27: $o,V3y: $o,V4y_27: $o] :
( ( ( V0c
=> ( V2x_27
=> V1x ) )
& ( ( (~) @ V0c )
=> ( V4y_27
=> V3y ) ) )
=> ( ( c_2Ebool_2ECOND @ $o @ V0c @ V2x_27 @ V4y_27 )
=> ( c_2Ebool_2ECOND @ $o @ V0c @ V1x @ V3y ) ) ) ).
thf(thm_2EConseqConv_2ECOND__CLAUSES__CT,axiom,
! [A_27a: $tType,V0t1: A_27a,V1t2: A_27a] :
( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2ET @ V0t1 @ V1t2 )
= V0t1 ) ).
thf(thm_2EConseqConv_2ECOND__CLAUSES__CF,axiom,
! [A_27a: $tType,V0t1: A_27a,V1t2: A_27a] :
( ( c_2Ebool_2ECOND @ A_27a @ c_2Ebool_2EF @ V0t1 @ V1t2 )
= V1t2 ) ).
thf(thm_2EConseqConv_2ECOND__CLAUSES__ID,axiom,
! [A_27a: $tType,V0b: $o,V1t: A_27a] :
( ( c_2Ebool_2ECOND @ A_27a @ V0b @ V1t @ V1t )
= V1t ) ).
thf(thm_2EConseqConv_2ECOND__CLAUSES__TT,axiom,
! [V0c: $o,V1x: $o] :
( ( c_2Ebool_2ECOND @ $o @ V0c @ c_2Ebool_2ET @ V1x )
<=> ( ( (~) @ V0c )
=> V1x ) ) ).
thf(thm_2EConseqConv_2ECOND__CLAUSES__FT,axiom,
! [V0c: $o,V1x: $o] :
( ( c_2Ebool_2ECOND @ $o @ V0c @ V1x @ c_2Ebool_2ET )
<=> ( V0c
=> V1x ) ) ).
thf(thm_2EConseqConv_2ECOND__CLAUSES__TF,axiom,
! [V0c: $o,V1x: $o] :
( ( c_2Ebool_2ECOND @ $o @ V0c @ c_2Ebool_2EF @ V1x )
<=> ( ( (~) @ V0c )
& V1x ) ) ).
thf(thm_2EConseqConv_2ECOND__CLAUSES__FF,axiom,
! [V0c: $o,V1x: $o] :
( ( c_2Ebool_2ECOND @ $o @ V0c @ V1x @ c_2Ebool_2EF )
<=> ( V0c
& V1x ) ) ).
thf(thm_2EConseqConv_2EASM__MARKER__THM,axiom,
! [V0y: $o,V1x: $o] :
( ( c_2EConseqConv_2EASM__MARKER @ V0y @ V1x )
= V1x ) ).
%------------------------------------------------------------------------------