ITP001 Axioms: ITP058^7.ax
%------------------------------------------------------------------------------
% File : ITP058^7 : 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 : wot.ax [Gau19]
% : HL4058^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 56 ( 10 unt; 36 typ; 0 def)
% Number of atoms : 59 ( 7 equ; 2 cnn)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 166 ( 2 ~; 2 |; 8 &; 134 @)
% ( 14 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg; 134 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 129 ( 129 >; 0 *; 0 +; 0 <<)
% Number of symbols : 37 ( 35 usr; 1 con; 0-4 aty)
% Number of variables : 82 ( 1 ^ 47 !; 3 ?; 82 :)
% ( 31 !>; 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(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
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_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Epred__set_2ECOMPL,type,
c_2Epred__set_2ECOMPL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Erelation_2ESTRORD,type,
c_2Erelation_2ESTRORD:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EStrongLinearOrder,type,
c_2Erelation_2EStrongLinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ewot_2EStrongWellOrder,type,
c_2Ewot_2EStrongWellOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWeakOrder,type,
c_2Erelation_2EWeakOrder:
!>[A_27g: $tType] : ( ( A_27g > A_27g > $o ) > $o ) ).
thf(c_2Ewot_2EWeakWellOrder,type,
c_2Ewot_2EWeakWellOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ewot_2Echain,type,
c_2Ewot_2Echain:
!>[A_27x: $tType] : ( ( ( A_27x > $o ) > $o ) > $o ) ).
thf(c_2Ewot_2Ecpl,type,
c_2Ewot_2Ecpl:
!>[A_27x: $tType] : ( ( A_27x > $o ) > ( A_27x > $o ) > $o ) ).
thf(c_2Ewot_2Emex,type,
c_2Ewot_2Emex:
!>[A_27x: $tType] : ( ( A_27x > $o ) > A_27x ) ).
thf(c_2Ewot_2Emex__less,type,
c_2Ewot_2Emex__less:
!>[A_27x: $tType] : ( A_27x > A_27x > $o ) ).
thf(c_2Ewot_2Emex__less__eq,type,
c_2Ewot_2Emex__less__eq:
!>[A_27x: $tType] : ( A_27x > A_27x > $o ) ).
thf(c_2Ewot_2Epreds,type,
c_2Ewot_2Epreds:
!>[A_27x: $tType] : ( A_27x > A_27x > $o ) ).
thf(c_2Ewot_2Epreds__image,type,
c_2Ewot_2Epreds__image:
!>[A_27x: $tType] : ( ( A_27x > $o ) > ( A_27x > $o ) > $o ) ).
thf(c_2Ewot_2Esetsuc,type,
c_2Ewot_2Esetsuc:
!>[A_27x: $tType] : ( ( A_27x > $o ) > A_27x > $o ) ).
thf(c_2Ewot_2Esuccl,type,
c_2Ewot_2Esuccl:
!>[A_27x: $tType] : ( ( ( A_27x > $o ) > $o ) > $o ) ).
thf(c_2Ewot_2Etower,type,
c_2Ewot_2Etower:
!>[A_27x: $tType] : ( ( ( A_27x > $o ) > $o ) > $o ) ).
thf(c_2Ewot_2Euncl,type,
c_2Ewot_2Euncl:
!>[A_27x: $tType] : ( ( ( A_27x > $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_2Ewot_2Ecpl__def,axiom,
! [A_27x: $tType,V0A: A_27x > $o,V1B: A_27x > $o] :
( ( c_2Ewot_2Ecpl @ A_27x @ V0A @ V1B )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27x @ V0A @ V1B )
| ( c_2Epred__set_2ESUBSET @ A_27x @ V1B @ V0A ) ) ) ).
thf(thm_2Ewot_2Echain__def,axiom,
! [A_27x: $tType,V0C: ( A_27x > $o ) > $o] :
( ( c_2Ewot_2Echain @ A_27x @ V0C )
<=> ! [V1a: A_27x > $o,V2b: A_27x > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27x > $o ) @ V1a @ V0C )
& ( c_2Ebool_2EIN @ ( A_27x > $o ) @ V2b @ V0C ) )
=> ( c_2Ewot_2Ecpl @ A_27x @ V1a @ V2b ) ) ) ).
thf(thm_2Ewot_2Emex__def,axiom,
! [A_27x: $tType,V0s: A_27x > $o] :
( ( c_2Ewot_2Emex @ A_27x @ V0s )
= ( c_2Epred__set_2ECHOICE @ A_27x @ ( c_2Epred__set_2ECOMPL @ A_27x @ V0s ) ) ) ).
thf(thm_2Ewot_2Esetsuc__def,axiom,
! [A_27x: $tType,V0s: A_27x > $o] :
( ( c_2Ewot_2Esetsuc @ A_27x @ V0s )
= ( c_2Epred__set_2EINSERT @ A_27x @ ( c_2Ewot_2Emex @ A_27x @ V0s ) @ V0s ) ) ).
thf(thm_2Ewot_2Esuccl__def,axiom,
! [A_27x: $tType,V0c: ( A_27x > $o ) > $o] :
( ( c_2Ewot_2Esuccl @ A_27x @ V0c )
<=> ! [V1s: A_27x > $o] :
( ( c_2Ebool_2EIN @ ( A_27x > $o ) @ V1s @ V0c )
=> ( c_2Ebool_2EIN @ ( A_27x > $o ) @ ( c_2Ewot_2Esetsuc @ A_27x @ V1s ) @ V0c ) ) ) ).
thf(thm_2Ewot_2Euncl__def,axiom,
! [A_27x: $tType,V0c: ( A_27x > $o ) > $o] :
( ( c_2Ewot_2Euncl @ A_27x @ V0c )
<=> ! [V1w: ( A_27x > $o ) > $o] :
( ( ( c_2Epred__set_2ESUBSET @ ( A_27x > $o ) @ V1w @ V0c )
& ( c_2Ewot_2Echain @ A_27x @ V1w ) )
=> ( c_2Ebool_2EIN @ ( A_27x > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27x @ V1w ) @ V0c ) ) ) ).
thf(thm_2Ewot_2Etower__def,axiom,
! [A_27x: $tType,V0A: ( A_27x > $o ) > $o] :
( ( c_2Ewot_2Etower @ A_27x @ V0A )
<=> ( ( c_2Ewot_2Esuccl @ A_27x @ V0A )
& ( c_2Ewot_2Euncl @ A_27x @ V0A ) ) ) ).
thf(thm_2Ewot_2Emex__less__eq__def,axiom,
! [A_27x: $tType,V0a: A_27x,V1b: A_27x] :
( ( c_2Ewot_2Emex__less__eq @ A_27x @ V0a @ V1b )
= ( c_2Epred__set_2ESUBSET @ A_27x @ ( c_2Ewot_2Epreds @ A_27x @ V0a ) @ ( c_2Ewot_2Epreds @ A_27x @ V1b ) ) ) ).
thf(thm_2Ewot_2Emex__less__def,axiom,
! [A_27x: $tType] :
( ( c_2Ewot_2Emex__less @ A_27x )
= ( c_2Erelation_2ESTRORD @ A_27x @ ( c_2Ewot_2Emex__less__eq @ A_27x ) ) ) ).
thf(thm_2Ewot_2EWeakWellOrder__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Ewot_2EWeakWellOrder @ A_27a @ V0R )
<=> ( ( c_2Erelation_2EWeakOrder @ A_27a @ V0R )
& ! [V1B: A_27a > $o] :
( ( (~)
@ ( V1B
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ? [V2m: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2m @ V1B )
& ! [V3b: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3b @ V1B )
=> ( V0R @ V2m @ V3b ) ) ) ) ) ) ).
thf(thm_2Ewot_2Epreds__image__def,axiom,
! [A_27x: $tType,V0X: A_27x > $o] :
( ( c_2Ewot_2Epreds__image @ A_27x @ V0X )
= ( c_2Epred__set_2EGSPEC @ ( A_27x > $o ) @ A_27x
@ ^ [V1x: A_27x] : ( c_2Epair_2E_2C @ ( A_27x > $o ) @ $o @ ( c_2Ewot_2Epreds @ A_27x @ V1x ) @ ( c_2Ebool_2EIN @ A_27x @ V1x @ V0X ) ) ) ) ).
thf(thm_2Ewot_2EStrongWellOrder__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Ewot_2EStrongWellOrder @ A_27a @ V0R )
<=> ( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0R )
& ( c_2Erelation_2EWF @ A_27a @ V0R ) ) ) ).
thf(thm_2Ewot_2EStrongWellOrderExists,axiom,
! [A_27a: $tType] :
? [V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EStrongLinearOrder @ A_27a @ V0R )
& ( c_2Erelation_2EWF @ A_27a @ V0R ) ) ).
%------------------------------------------------------------------------------