ITP001 Axioms: ITP019^7.ax
%------------------------------------------------------------------------------
% File : ITP019^7 : 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 : while.ax [Gau19]
% : HL4019^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 61 ( 17 unt; 28 typ; 0 def)
% Number of atoms : 67 ( 22 equ; 11 cnn)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 388 ( 11 ~; 1 |; 19 &; 312 @)
% ( 12 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg; 312 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 107 ( 107 >; 0 *; 0 +; 0 <<)
% Number of symbols : 28 ( 26 usr; 4 con; 0-6 aty)
% Number of variables : 143 ( 13 ^ 110 !; 5 ?; 143 :)
% ( 15 !>; 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_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $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_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $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_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_2Earithmetic_2EFUNPOW,type,
c_2Earithmetic_2EFUNPOW:
!>[A_27a: $tType] : ( ( A_27a > A_27a ) > tyop_2Enum_2Enum > A_27a > A_27a ) ).
thf(c_2Ewhile_2EHOARE__SPEC,type,
c_2Ewhile_2EHOARE__SPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > $o ) > $o ) ).
thf(c_2Ewhile_2ELEAST,type,
c_2Ewhile_2ELEAST: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Ewhile_2EOLEAST,type,
c_2Ewhile_2EOLEAST: ( tyop_2Enum_2Enum > $o ) > ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) ).
thf(c_2Ewhile_2EOWHILE,type,
c_2Ewhile_2EOWHILE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > A_27a ) > A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ewhile_2EWHILE,type,
c_2Ewhile_2EWHILE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > A_27a ) > A_27a > A_27a ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
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_2Ewhile_2EWHILE,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1g: A_27a > A_27a,V2x: A_27a] :
( ( c_2Ewhile_2EWHILE @ A_27a @ V0P @ V1g @ V2x )
= ( c_2Ebool_2ECOND @ A_27a @ ( V0P @ V2x ) @ ( c_2Ewhile_2EWHILE @ A_27a @ V0P @ V1g @ ( V1g @ V2x ) ) @ V2x ) ) ).
thf(thm_2Ewhile_2EHOARE__SPEC__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1C: A_27a > A_27b,V2Q: A_27b > $o] :
( ( c_2Ewhile_2EHOARE__SPEC @ A_27a @ A_27b @ V0P @ V1C @ V2Q )
<=> ! [V3s: A_27a] :
( ( V0P @ V3s )
=> ( V2Q @ ( V1C @ V3s ) ) ) ) ).
thf(thm_2Ewhile_2ELEAST__DEF,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ( c_2Ewhile_2ELEAST @ V0P )
= ( c_2Ewhile_2EWHILE @ tyop_2Enum_2Enum @ ( c_2Ecombin_2Eo @ tyop_2Enum_2Enum @ $o @ $o @ c_2Ebool_2E_7E @ V0P ) @ c_2Enum_2ESUC @ c_2Enum_2E0 ) ) ).
thf(thm_2Ewhile_2EOLEAST__def,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ( c_2Ewhile_2EOLEAST @ V0P )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum )
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( V0P @ V1n ) )
@ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum
@ ( c_2Ewhile_2ELEAST
@ ^ [V2n: tyop_2Enum_2Enum] : ( V0P @ V2n ) ) )
@ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Ewhile_2EOWHILE__def,axiom,
! [A_27a: $tType,V0G: A_27a > $o,V1f: A_27a > A_27a,V2s: A_27a] :
( ( c_2Ewhile_2EOWHILE @ A_27a @ V0G @ V1f @ V2s )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a )
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Ebool_2E_7E @ ( V0G @ ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f @ V3n @ V2s ) ) ) )
@ ( c_2Eoption_2ESOME @ A_27a
@ ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f
@ ( c_2Ewhile_2ELEAST
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2E_7E @ ( V0G @ ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f @ V4n @ V2s ) ) ) )
@ V2s ) )
@ ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Ewhile_2EITERATION,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1g: A_27a > A_27a] :
? [V2f: A_27a > A_27a] :
! [V3x: A_27a] :
( ( V2f @ V3x )
= ( c_2Ebool_2ECOND @ A_27a @ ( V0P @ V3x ) @ V3x @ ( V2f @ ( V1g @ V3x ) ) ) ) ).
thf(thm_2Ewhile_2EWHILE__INDUCTION,axiom,
! [A_27a: $tType,V0B: A_27a > $o,V1C: A_27a > A_27a,V2R: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2EWF @ A_27a @ V2R )
& ! [V3s: A_27a] :
( ( V0B @ V3s )
=> ( V2R @ ( V1C @ V3s ) @ V3s ) ) )
=> ! [V4P: A_27a > $o] :
( ! [V5s: A_27a] :
( ( ( V0B @ V5s )
=> ( V4P @ ( V1C @ V5s ) ) )
=> ( V4P @ V5s ) )
=> ! [V6v: A_27a] : ( V4P @ V6v ) ) ) ).
thf(thm_2Ewhile_2EWHILE__RULE,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1R: A_27a > A_27a > $o,V2B: A_27a > $o,V3C: A_27a > A_27a] :
( ( ( c_2Erelation_2EWF @ A_27a @ V1R )
& ! [V4s: A_27a] :
( ( V2B @ V4s )
=> ( V1R @ ( V3C @ V4s ) @ V4s ) ) )
=> ( ( c_2Ewhile_2EHOARE__SPEC @ A_27a @ A_27a
@ ^ [V5s: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V5s ) @ ( V2B @ V5s ) )
@ V3C
@ V0P )
=> ( c_2Ewhile_2EHOARE__SPEC @ A_27a @ A_27a @ V0P @ ( c_2Ewhile_2EWHILE @ A_27a @ V2B @ V3C )
@ ^ [V6s: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V0P @ V6s ) @ ( c_2Ebool_2E_7E @ ( V2B @ V6s ) ) ) ) ) ) ).
thf(thm_2Ewhile_2ELEAST__INTRO,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum] :
( ( V0P @ V1x )
=> ( V0P @ ( c_2Ewhile_2ELEAST @ V0P ) ) ) ).
thf(thm_2Ewhile_2ELESS__LEAST,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1m @ ( c_2Ewhile_2ELEAST @ V0P ) )
=> ( (~) @ ( V0P @ V1m ) ) ) ).
thf(thm_2Ewhile_2EFULL__LEAST__INTRO,axiom,
! [V0P: tyop_2Enum_2Enum > $o,V1x: tyop_2Enum_2Enum] :
( ( V0P @ V1x )
=> ( ( V0P @ ( c_2Ewhile_2ELEAST @ V0P ) )
& ( c_2Earithmetic_2E_3C_3D @ ( c_2Ewhile_2ELEAST @ V0P ) @ V1x ) ) ) ).
thf(thm_2Ewhile_2ELEAST__ELIM,axiom,
! [V0Q: tyop_2Enum_2Enum > $o,V1P: tyop_2Enum_2Enum > $o] :
( ( ? [V2n: tyop_2Enum_2Enum] : ( V1P @ V2n )
& ! [V3n: tyop_2Enum_2Enum] :
( ( ! [V4m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V4m @ V3n )
=> ( (~) @ ( V1P @ V4m ) ) )
& ( V1P @ V3n ) )
=> ( V0Q @ V3n ) ) )
=> ( V0Q @ ( c_2Ewhile_2ELEAST @ V1P ) ) ) ).
thf(thm_2Ewhile_2ELEAST__EXISTS,axiom,
! [V0p: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( V0p @ V1n )
<=> ( ( V0p @ ( c_2Ewhile_2ELEAST @ V0p ) )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Ewhile_2ELEAST @ V0p ) )
=> ( (~) @ ( V0p @ V2n ) ) ) ) ) ).
thf(thm_2Ewhile_2ELEAST__EXISTS__IMP,axiom,
! [V0p: tyop_2Enum_2Enum > $o] :
( ? [V1n: tyop_2Enum_2Enum] : ( V0p @ V1n )
=> ( ( V0p @ ( c_2Ewhile_2ELEAST @ V0p ) )
& ! [V2n: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Ewhile_2ELEAST @ V0p ) )
=> ( (~) @ ( V0p @ V2n ) ) ) ) ) ).
thf(thm_2Ewhile_2ELEAST__EQ,axiom,
! [V0x: tyop_2Enum_2Enum] :
( ( ( c_2Ewhile_2ELEAST
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1n @ V0x ) )
= V0x )
& ( ( c_2Ewhile_2ELEAST
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0x @ V2n ) )
= V0x ) ) ).
thf(thm_2Ewhile_2ELEAST__T,axiom,
( ( c_2Ewhile_2ELEAST
@ ^ [V0x: tyop_2Enum_2Enum] : c_2Ebool_2ET )
= c_2Enum_2E0 ) ).
thf(thm_2Ewhile_2EOLEAST__INTRO,axiom,
! [V0Q: ( tyop_2Eoption_2Eoption @ tyop_2Enum_2Enum ) > $o,V1P: tyop_2Enum_2Enum > $o] :
( ( ( ! [V2n: tyop_2Enum_2Enum] : ( (~) @ ( V1P @ V2n ) )
=> ( V0Q @ ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) ) )
& ! [V3n: tyop_2Enum_2Enum] :
( ( ( V1P @ V3n )
& ! [V4m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V4m @ V3n )
=> ( (~) @ ( V1P @ V4m ) ) ) )
=> ( V0Q @ ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V3n ) ) ) )
=> ( V0Q @ ( c_2Ewhile_2EOLEAST @ V1P ) ) ) ).
thf(thm_2Ewhile_2EOLEAST__EQNS,axiom,
! [V0x: tyop_2Enum_2Enum] :
( ( ( c_2Ewhile_2EOLEAST
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1n @ V0x ) )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V0x ) )
& ( ( c_2Ewhile_2EOLEAST
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0x @ V2n ) )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V0x ) )
& ( ( c_2Ewhile_2EOLEAST
@ ^ [V3n: tyop_2Enum_2Enum] : c_2Ebool_2EF )
= ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) )
& ( ( c_2Ewhile_2EOLEAST
@ ^ [V4n: tyop_2Enum_2Enum] : c_2Ebool_2ET )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ewhile_2EOLEAST__EQ__NONE,axiom,
! [V0P: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ewhile_2EOLEAST @ V0P )
= ( c_2Eoption_2ENONE @ tyop_2Enum_2Enum ) )
<=> ! [V1n: tyop_2Enum_2Enum] : ( (~) @ ( V0P @ V1n ) ) ) ).
thf(thm_2Ewhile_2EOLEAST__EQ__SOME,axiom,
! [V0n: tyop_2Enum_2Enum,V1P: tyop_2Enum_2Enum > $o] :
( ( ( c_2Ewhile_2EOLEAST @ V1P )
= ( c_2Eoption_2ESOME @ tyop_2Enum_2Enum @ V0n ) )
<=> ( ( V1P @ V0n )
& ! [V2m: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2m @ V0n )
=> ( (~) @ ( V1P @ V2m ) ) ) ) ) ).
thf(thm_2Ewhile_2EOWHILE__THM,axiom,
! [A_27a: $tType,V0s: A_27a,V1f: A_27a > A_27a,V2G: A_27a > $o] :
( ( c_2Ewhile_2EOWHILE @ A_27a @ V2G @ V1f @ V0s )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V2G @ V0s ) @ ( c_2Ewhile_2EOWHILE @ A_27a @ V2G @ V1f @ ( V1f @ V0s ) ) @ ( c_2Eoption_2ESOME @ A_27a @ V0s ) ) ) ).
thf(thm_2Ewhile_2EOWHILE__EQ__NONE,axiom,
! [A_27a: $tType,V0s: A_27a,V1f: A_27a > A_27a,V2G: A_27a > $o] :
( ( ( c_2Ewhile_2EOWHILE @ A_27a @ V2G @ V1f @ V0s )
= ( c_2Eoption_2ENONE @ A_27a ) )
<=> ! [V3n: tyop_2Enum_2Enum] : ( V2G @ ( c_2Earithmetic_2EFUNPOW @ A_27a @ V1f @ V3n @ V0s ) ) ) ).
thf(thm_2Ewhile_2EOWHILE__ENDCOND,axiom,
! [A_27a: $tType,V0s_27: A_27a,V1s: A_27a,V2f: A_27a > A_27a,V3G: A_27a > $o] :
( ( ( c_2Ewhile_2EOWHILE @ A_27a @ V3G @ V2f @ V1s )
= ( c_2Eoption_2ESOME @ A_27a @ V0s_27 ) )
=> ( (~) @ ( V3G @ V0s_27 ) ) ) ).
thf(thm_2Ewhile_2EOWHILE__WHILE,axiom,
! [A_27a: $tType,V0s_27: A_27a,V1s: A_27a,V2f: A_27a > A_27a,V3G: A_27a > $o] :
( ( ( c_2Ewhile_2EOWHILE @ A_27a @ V3G @ V2f @ V1s )
= ( c_2Eoption_2ESOME @ A_27a @ V0s_27 ) )
=> ( ( c_2Ewhile_2EWHILE @ A_27a @ V3G @ V2f @ V1s )
= V0s_27 ) ) ).
thf(thm_2Ewhile_2EOWHILE__INV__IND,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1G: A_27a > $o,V2f: A_27a > A_27a,V3s: A_27a] :
( ( ( V0P @ V3s )
& ! [V4x: A_27a] :
( ( ( V0P @ V4x )
& ( V1G @ V4x ) )
=> ( V0P @ ( V2f @ V4x ) ) ) )
=> ! [V5s_27: A_27a] :
( ( ( c_2Ewhile_2EOWHILE @ A_27a @ V1G @ V2f @ V3s )
= ( c_2Eoption_2ESOME @ A_27a @ V5s_27 ) )
=> ( V0P @ V5s_27 ) ) ) ).
thf(thm_2Ewhile_2EOWHILE__IND,axiom,
! [A_27a: $tType,V0P: A_27a > A_27a > $o,V1G: A_27a > $o,V2f: A_27a > A_27a] :
( ( ! [V3s: A_27a] :
( ( (~) @ ( V1G @ V3s ) )
=> ( V0P @ V3s @ V3s ) )
& ! [V4s1: A_27a,V5s2: A_27a] :
( ( ( V1G @ V4s1 )
& ( V0P @ ( V2f @ V4s1 ) @ V5s2 ) )
=> ( V0P @ V4s1 @ V5s2 ) ) )
=> ! [V6s1: A_27a,V7s2: A_27a] :
( ( ( c_2Ewhile_2EOWHILE @ A_27a @ V1G @ V2f @ V6s1 )
= ( c_2Eoption_2ESOME @ A_27a @ V7s2 ) )
=> ( V0P @ V6s1 @ V7s2 ) ) ) ).
%------------------------------------------------------------------------------