TPTP Problem File: ITP002^7.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP002^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic export of thm_2Eoption_2EOPTION__MAP2__THM.p, 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 : thm_2Eoption_2EOPTION__MAP2__THM.p [Gau20]
% : HL400501^7.p [TPAP]
% Status : Theorem
% Rating : 0.67 v8.1.0, 1.00 v7.5.0
% Syntax : Number of formulae : 1011 ( 298 unt; 289 typ; 0 def)
% Number of atoms : 1598 ( 448 equ; 151 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 6492 ( 151 ~; 162 |; 396 &;4905 @)
% ( 352 <=>; 526 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 1678 (1678 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 140 usr; 7 con; 0-7 aty)
% Number of variables : 2981 ( 145 ^;2478 !; 105 ?;2981 :)
% ( 253 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP002^7.ax').
include('Axioms/ITP001/ITP003^7.ax').
include('Axioms/ITP001/ITP004^7.ax').
include('Axioms/ITP001/ITP005^7.ax').
include('Axioms/ITP001/ITP006^7.ax').
include('Axioms/ITP001/ITP007^7.ax').
include('Axioms/ITP001/ITP008^7.ax').
include('Axioms/ITP001/ITP009^7.ax').
include('Axioms/ITP001/ITP010^7.ax').
include('Axioms/ITP001/ITP011^7.ax').
include('Axioms/ITP001/ITP012^7.ax').
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Eone_2Eone,type,
tyop_2Eone_2Eone: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $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_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_2Esum_2EINL,type,
c_2Esum_2EINL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Esum_2EINR,type,
c_2Esum_2EINR:
!>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).
thf(c_2Eoption_2EIS__NONE,type,
c_2Eoption_2EIS__NONE:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > $o ) ).
thf(c_2Eoption_2EIS__SOME,type,
c_2Eoption_2EIS__SOME:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > $o ) ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Eoption_2EOPTION__JOIN,type,
c_2Eoption_2EOPTION__JOIN:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ ( tyop_2Eoption_2Eoption @ A_27a ) ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Eoption_2EOPTION__MAP,type,
c_2Eoption_2EOPTION__MAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) ) ).
thf(c_2Eoption_2EOPTION__MAP2,type,
c_2Eoption_2EOPTION__MAP2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > A_27c > A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) > ( tyop_2Eoption_2Eoption @ A_27c ) > ( 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_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Eoption_2ETHE,type,
c_2Eoption_2ETHE:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27a ) ).
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_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Eone_2Eone,type,
c_2Eone_2Eone: tyop_2Eone_2Eone ).
thf(c_2Eoption_2Eoption__ABS,type,
c_2Eoption_2Eoption__ABS:
!>[A_27a: $tType] : ( ( tyop_2Esum_2Esum @ A_27a @ tyop_2Eone_2Eone ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Eoption_2Eoption__CASE,type,
c_2Eoption_2Eoption__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27b > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Eoption_2Eoption__REP,type,
c_2Eoption_2Eoption__REP:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Esum_2Esum @ A_27a @ tyop_2Eone_2Eone ) ) ).
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_2Eoption_2Eoption__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Esum_2Esum @ A_27a @ tyop_2Eone_2Eone )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Esum_2Esum @ A_27a @ tyop_2Eone_2Eone ) @ ( tyop_2Eoption_2Eoption @ A_27a )
@ ^ [V1x: tyop_2Esum_2Esum @ A_27a @ tyop_2Eone_2Eone] : c_2Ebool_2ET
@ V0rep ) ).
thf(thm_2Eoption_2Eoption__REP__ABS__DEF,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2Eoption__ABS @ A_27a @ ( c_2Eoption_2Eoption__REP @ A_27a @ V0a ) )
= V0a )
& ! [V1r: tyop_2Esum_2Esum @ A_27a @ tyop_2Eone_2Eone] :
( ( ^ [V2x: tyop_2Esum_2Esum @ A_27a @ tyop_2Eone_2Eone] : c_2Ebool_2ET
@ V1r )
<=> ( ( c_2Eoption_2Eoption__REP @ A_27a @ ( c_2Eoption_2Eoption__ABS @ A_27a @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Eoption_2ESOME__DEF,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Eoption_2ESOME @ A_27a @ V0x )
= ( c_2Eoption_2Eoption__ABS @ A_27a @ ( c_2Esum_2EINL @ A_27a @ tyop_2Eone_2Eone @ V0x ) ) ) ).
thf(thm_2Eoption_2ENONE__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Eoption_2ENONE @ A_27a )
= ( c_2Eoption_2Eoption__ABS @ A_27a @ ( c_2Esum_2EINR @ A_27a @ tyop_2Eone_2Eone @ c_2Eone_2Eone ) ) ) ).
thf(thm_2Eoption_2Eoption__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0v: A_27b,V1f: A_27a > A_27b] :
( ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b @ ( c_2Eoption_2ENONE @ A_27a ) @ V0v @ V1f )
= V0v )
& ! [V2x: A_27a,V3v: A_27b,V4f: A_27a > A_27b] :
( ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b @ ( c_2Eoption_2ESOME @ A_27a @ V2x ) @ V3v @ V4f )
= ( V4f @ V2x ) ) ) ).
thf(thm_2Eoption_2EOPTION__MAP__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0f: A_27a > A_27b,V1x: A_27a] :
( ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V0f @ ( c_2Eoption_2ESOME @ A_27a @ V1x ) )
= ( c_2Eoption_2ESOME @ A_27b @ ( V0f @ V1x ) ) )
& ! [V2f: A_27a > A_27b] :
( ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V2f @ ( c_2Eoption_2ENONE @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27b ) ) ) ).
thf(thm_2Eoption_2EIS__SOME__DEF,axiom,
! [A_27a: $tType] :
( ! [V0x: A_27a] :
( ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Eoption_2ESOME @ A_27a @ V0x ) )
= c_2Ebool_2ET )
& ( ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Eoption_2ENONE @ A_27a ) )
= c_2Ebool_2EF ) ) ).
thf(thm_2Eoption_2EIS__NONE__DEF,axiom,
! [A_27a: $tType] :
( ! [V0x: A_27a] :
( ( c_2Eoption_2EIS__NONE @ A_27a @ ( c_2Eoption_2ESOME @ A_27a @ V0x ) )
= c_2Ebool_2EF )
& ( ( c_2Eoption_2EIS__NONE @ A_27a @ ( c_2Eoption_2ENONE @ A_27a ) )
= c_2Ebool_2ET ) ) ).
thf(thm_2Eoption_2ETHE__DEF,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( c_2Eoption_2ETHE @ A_27a @ ( c_2Eoption_2ESOME @ A_27a @ V0x ) )
= V0x ) ).
thf(thm_2Eoption_2EOPTION__MAP2__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > A_27c > A_27a,V1x: tyop_2Eoption_2Eoption @ A_27b,V2y: tyop_2Eoption_2Eoption @ A_27c] :
( ( c_2Eoption_2EOPTION__MAP2 @ A_27a @ A_27b @ A_27c @ V0f @ V1x @ V2y )
= ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Eoption_2EIS__SOME @ A_27b @ V1x ) @ ( c_2Eoption_2EIS__SOME @ A_27c @ V2y ) ) @ ( c_2Eoption_2ESOME @ A_27a @ ( V0f @ ( c_2Eoption_2ETHE @ A_27b @ V1x ) @ ( c_2Eoption_2ETHE @ A_27c @ V2y ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Eoption_2EOPTION__JOIN__DEF,axiom,
! [A_27a: $tType] :
( ( ( c_2Eoption_2EOPTION__JOIN @ A_27a @ ( c_2Eoption_2ENONE @ ( tyop_2Eoption_2Eoption @ A_27a ) ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V0x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EOPTION__JOIN @ A_27a @ ( c_2Eoption_2ESOME @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V0x ) )
= V0x ) ) ).
thf(thm_2Eoption_2Eoption__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0e: A_27b,V1f: A_27a > A_27b] :
? [V2fn: ( tyop_2Eoption_2Eoption @ A_27a ) > A_27b] :
( ( ( V2fn @ ( c_2Eoption_2ENONE @ A_27a ) )
= V0e )
& ! [V3x: A_27a] :
( ( V2fn @ ( c_2Eoption_2ESOME @ A_27a @ V3x ) )
= ( V1f @ V3x ) ) ) ).
thf(thm_2Eoption_2Eoption__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eoption_2Eoption @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V1a: A_27a] : ( V0P @ ( c_2Eoption_2ESOME @ A_27a @ V1a ) ) )
=> ! [V2x: tyop_2Eoption_2Eoption @ A_27a] : ( V0P @ V2x ) ) ).
thf(thm_2Eoption_2Eoption__nchotomy,axiom,
! [A_27a: $tType,V0opt: tyop_2Eoption_2Eoption @ A_27a] :
( ( V0opt
= ( c_2Eoption_2ENONE @ A_27a ) )
| ? [V1x: A_27a] :
( V0opt
= ( c_2Eoption_2ESOME @ A_27a @ V1x ) ) ) ).
thf(thm_2Eoption_2EFORALL__OPTION,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eoption_2Eoption @ A_27a ) > $o] :
( ! [V1opt: tyop_2Eoption_2Eoption @ A_27a] : ( V0P @ V1opt )
<=> ( ( V0P @ ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V2x: A_27a] : ( V0P @ ( c_2Eoption_2ESOME @ A_27a @ V2x ) ) ) ) ).
thf(thm_2Eoption_2EEXISTS__OPTION,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eoption_2Eoption @ A_27a ) > $o] :
( ? [V1opt: tyop_2Eoption_2Eoption @ A_27a] : ( V0P @ V1opt )
<=> ( ( V0P @ ( c_2Eoption_2ENONE @ A_27a ) )
| ? [V2x: A_27a] : ( V0P @ ( c_2Eoption_2ESOME @ A_27a @ V2x ) ) ) ) ).
thf(thm_2Eoption_2ESOME__11,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] :
( ( ( c_2Eoption_2ESOME @ A_27a @ V0x )
= ( c_2Eoption_2ESOME @ A_27a @ V1y ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Eoption_2ENOT__NONE__SOME,axiom,
! [A_27a: $tType,V0x: A_27a] :
( (~)
@ ( ( c_2Eoption_2ENONE @ A_27a )
= ( c_2Eoption_2ESOME @ A_27a @ V0x ) ) ) ).
thf(thm_2Eoption_2ENOT__SOME__NONE,axiom,
! [A_27a: $tType,V0x: A_27a] :
( (~)
@ ( ( c_2Eoption_2ESOME @ A_27a @ V0x )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Eoption_2EOPTION__MAP2__THM,conjecture,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0y: A_27c,V1x: A_27b,V2f: A_27b > A_27c > A_27a] :
( ( ( c_2Eoption_2EOPTION__MAP2 @ A_27a @ A_27b @ A_27c @ V2f @ ( c_2Eoption_2ESOME @ A_27b @ V1x ) @ ( c_2Eoption_2ESOME @ A_27c @ V0y ) )
= ( c_2Eoption_2ESOME @ A_27a @ ( V2f @ V1x @ V0y ) ) )
& ( ( c_2Eoption_2EOPTION__MAP2 @ A_27a @ A_27b @ A_27c @ V2f @ ( c_2Eoption_2ESOME @ A_27b @ V1x ) @ ( c_2Eoption_2ENONE @ A_27c ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Eoption_2EOPTION__MAP2 @ A_27a @ A_27b @ A_27c @ V2f @ ( c_2Eoption_2ENONE @ A_27b ) @ ( c_2Eoption_2ESOME @ A_27c @ V0y ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( ( c_2Eoption_2EOPTION__MAP2 @ A_27a @ A_27b @ A_27c @ V2f @ ( c_2Eoption_2ENONE @ A_27b ) @ ( c_2Eoption_2ENONE @ A_27c ) )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
%------------------------------------------------------------------------------