ITP001 Axioms: ITP009^7.ax
%------------------------------------------------------------------------------
% File : ITP009^7 : 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 : num.ax [Gau19]
% : HL4009^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 38 ( 8 unt; 21 typ; 0 def)
% Number of atoms : 26 ( 9 equ; 4 cnn)
% Maximal formula atoms : 3 ( 0 avg)
% Number of connectives : 80 ( 4 ~; 1 |; 5 &; 55 @)
% ( 9 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg; 55 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 33 ( 33 >; 0 *; 0 +; 0 <<)
% Number of symbols : 20 ( 18 usr; 3 con; 0-4 aty)
% Number of variables : 39 ( 0 ^ 28 !; 2 ?; 39 :)
% ( 9 !>; 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_2Emin_2Eind,type,
tyop_2Emin_2Eind: $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $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_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
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_2Enum_2EABS__num,type,
c_2Enum_2EABS__num: tyop_2Emin_2Eind > tyop_2Enum_2Enum ).
thf(c_2Enum_2EIS__NUM__REP,type,
c_2Enum_2EIS__NUM__REP: tyop_2Emin_2Eind > $o ).
thf(c_2Ebool_2EONE__ONE,type,
c_2Ebool_2EONE__ONE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).
thf(c_2Ebool_2EONTO,type,
c_2Ebool_2EONTO:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).
thf(c_2Enum_2EREP__num,type,
c_2Enum_2EREP__num: tyop_2Enum_2Enum > tyop_2Emin_2Eind ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enum_2ESUC__REP,type,
c_2Enum_2ESUC__REP: tyop_2Emin_2Eind > tyop_2Emin_2Eind ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Enum_2EZERO__REP,type,
c_2Enum_2EZERO__REP: tyop_2Emin_2Eind ).
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_2Enum_2ESUC__REP__DEF,axiom,
( ( c_2Ebool_2EONE__ONE @ tyop_2Emin_2Eind @ tyop_2Emin_2Eind @ c_2Enum_2ESUC__REP )
& ( (~) @ ( c_2Ebool_2EONTO @ tyop_2Emin_2Eind @ tyop_2Emin_2Eind @ c_2Enum_2ESUC__REP ) ) ) ).
thf(thm_2Enum_2EZERO__REP__DEF,axiom,
! [V0y: tyop_2Emin_2Eind] :
( (~)
@ ( c_2Enum_2EZERO__REP
= ( c_2Enum_2ESUC__REP @ V0y ) ) ) ).
thf(thm_2Enum_2EIS__NUM__REP,axiom,
! [V0m: tyop_2Emin_2Eind] :
( ( c_2Enum_2EIS__NUM__REP @ V0m )
<=> ! [V1P: tyop_2Emin_2Eind > $o] :
( ( ( V1P @ c_2Enum_2EZERO__REP )
& ! [V2n: tyop_2Emin_2Eind] :
( ( V1P @ V2n )
=> ( V1P @ ( c_2Enum_2ESUC__REP @ V2n ) ) ) )
=> ( V1P @ V0m ) ) ) ).
thf(thm_2Enum_2Enum__TY__DEF,axiom,
? [V0rep: tyop_2Enum_2Enum > tyop_2Emin_2Eind] : ( c_2Ebool_2ETYPE__DEFINITION @ tyop_2Emin_2Eind @ tyop_2Enum_2Enum @ c_2Enum_2EIS__NUM__REP @ V0rep ) ).
thf(thm_2Enum_2Enum__ISO__DEF,axiom,
( ! [V0a: tyop_2Enum_2Enum] :
( ( c_2Enum_2EABS__num @ ( c_2Enum_2EREP__num @ V0a ) )
= V0a )
& ! [V1r: tyop_2Emin_2Eind] :
( ( c_2Enum_2EIS__NUM__REP @ V1r )
<=> ( ( c_2Enum_2EREP__num @ ( c_2Enum_2EABS__num @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Enum_2EZERO__DEF,axiom,
( c_2Enum_2E0
= ( c_2Enum_2EABS__num @ c_2Enum_2EZERO__REP ) ) ).
thf(thm_2Enum_2ESUC__DEF,axiom,
! [V0m: tyop_2Enum_2Enum] :
( ( c_2Enum_2ESUC @ V0m )
= ( c_2Enum_2EABS__num @ ( c_2Enum_2ESUC__REP @ ( c_2Enum_2EREP__num @ V0m ) ) ) ) ).
thf(thm_2Enum_2ENOT__SUC,axiom,
! [V0n: tyop_2Enum_2Enum] :
( (~)
@ ( ( c_2Enum_2ESUC @ V0n )
= c_2Enum_2E0 ) ) ).
thf(thm_2Enum_2EINV__SUC,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Enum_2ESUC @ V0m )
= ( c_2Enum_2ESUC @ V1n ) )
=> ( V0m = V1n ) ) ).
thf(thm_2Enum_2EINDUCTION,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ( ( V0P @ c_2Enum_2E0 )
& ! [V1n: tyop_2Enum_2Enum] :
( ( V0P @ V1n )
=> ( V0P @ ( c_2Enum_2ESUC @ V1n ) ) ) )
=> ! [V2n: tyop_2Enum_2Enum] : ( V0P @ V2n ) ) ).
%------------------------------------------------------------------------------