TPTP Problem File: ITP011^3.p
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%------------------------------------------------------------------------------
% File : ITP011^3 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 syntactic export of thm_2Equotient__option_2EOPTION__REL__def.p, bushy 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 : thm_2Equotient__option_2EOPTION__REL__def.p [Gau19]
% : HL405001^3.p [TPAP]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 40 ( 8 unt; 21 typ; 0 def)
% Number of atoms : 94 ( 36 equ; 10 cnn)
% Maximal formula atoms : 30 ( 4 avg)
% Number of connectives : 312 ( 10 ~; 7 |; 44 &; 211 @)
% ( 30 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 44 ( 44 >; 0 *; 0 +; 0 <<)
% Number of symbols : 22 ( 20 usr; 3 con; 0-5 aty)
% Number of variables : 82 ( 0 ^; 63 !; 4 ?; 82 :)
% ( 15 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $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_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_2EF,type,
c_2Ebool_2EF: $o ).
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_2EOPTREL,type,
c_2Eoption_2EOPTREL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Eoption_2Eoption @ A_27a ) > ( tyop_2Eoption_2Eoption @ A_27b ) > $o ) ).
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_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
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_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_2Ebool_2ETRUTH,axiom,
c_2Ebool_2ET ).
thf(thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: $o,V1t2: $o] :
( ( V0t1
=> V1t2 )
=> ( ( V1t2
=> V0t1 )
=> ( V0t1 = V1t2 ) ) ) ).
thf(thm_2Ebool_2EFALSITY,axiom,
! [V0t: $o] :
( c_2Ebool_2EF
=> V0t ) ).
thf(thm_2Ebool_2EEXISTS__SIMP,axiom,
! [A_27a: $tType,V0t: $o] :
( ? [V1x: A_27a] : V0t
<=> V0t ) ).
thf(thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
& V0t )
<=> V0t )
& ( ( V0t
& c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF
& V0t )
<=> c_2Ebool_2EF )
& ( ( V0t
& c_2Ebool_2EF )
<=> c_2Ebool_2EF )
& ( ( V0t
& V0t )
<=> V0t ) ) ).
thf(thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET
| V0t )
<=> c_2Ebool_2ET )
& ( ( V0t
| c_2Ebool_2ET )
<=> c_2Ebool_2ET )
& ( ( c_2Ebool_2EF
| V0t )
<=> V0t )
& ( ( V0t
| c_2Ebool_2EF )
<=> V0t )
& ( ( V0t
| V0t )
<=> V0t ) ) ).
thf(thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: $o] :
( ( (~) @ ( (~) @ V0t ) )
<=> V0t )
& ( ( (~) @ c_2Ebool_2ET )
<=> c_2Ebool_2EF )
& ( ( (~) @ c_2Ebool_2EF )
<=> c_2Ebool_2ET ) ) ).
thf(thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: $tType,V0x: A_27a] :
( ( V0x = V0x )
<=> c_2Ebool_2ET ) ).
thf(thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: $o] :
( ( ( c_2Ebool_2ET = V0t )
<=> V0t )
& ( ( V0t = c_2Ebool_2ET )
<=> V0t )
& ( ( c_2Ebool_2EF = V0t )
<=> ( (~) @ V0t ) )
& ( ( V0t = c_2Ebool_2EF )
<=> ( (~) @ V0t ) ) ) ).
thf(thm_2Eoption_2Eoption__CLAUSES,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1e: tyop_2Eoption_2Eoption @ A_27a,V2e: A_27b] :
( ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Eoption_2ESOME @ A_27a @ V3x )
= ( c_2Eoption_2ESOME @ A_27a @ V4y ) )
<=> ( V3x = V4y ) )
& ! [V5x: A_27a] :
( ( c_2Eoption_2ETHE @ A_27a @ ( c_2Eoption_2ESOME @ A_27a @ V5x ) )
= V5x )
& ! [V6x: A_27a] :
( (~)
@ ( ( c_2Eoption_2ENONE @ A_27a )
= ( c_2Eoption_2ESOME @ A_27a @ V6x ) ) )
& ! [V7x: A_27a] :
( (~)
@ ( ( c_2Eoption_2ESOME @ A_27a @ V7x )
= ( c_2Eoption_2ENONE @ A_27a ) ) )
& ! [V8x: A_27a] :
( ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Eoption_2ESOME @ A_27a @ V8x ) )
= c_2Ebool_2ET )
& ( ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Eoption_2ENONE @ A_27a ) )
= c_2Ebool_2EF )
& ! [V9x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EIS__NONE @ A_27a @ V9x )
<=> ( V9x
= ( c_2Eoption_2ENONE @ A_27a ) ) )
& ! [V10x: tyop_2Eoption_2Eoption @ A_27a] :
( ( (~) @ ( c_2Eoption_2EIS__SOME @ A_27a @ V10x ) )
<=> ( V10x
= ( c_2Eoption_2ENONE @ A_27a ) ) )
& ! [V11x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EIS__SOME @ A_27a @ V11x )
=> ( ( c_2Eoption_2ESOME @ A_27a @ ( c_2Eoption_2ETHE @ A_27a @ V11x ) )
= V11x ) )
& ! [V12x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V12x @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27a ) )
= V12x )
& ! [V13x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V13x @ V13x @ ( c_2Eoption_2ESOME @ A_27a ) )
= V13x )
& ! [V14x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EIS__NONE @ A_27a @ V14x )
=> ( ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b @ V14x @ V2e @ V0f )
= V2e ) )
& ! [V15x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EIS__SOME @ A_27a @ V15x )
=> ( ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b @ V15x @ V2e @ V0f )
= ( V0f @ ( c_2Eoption_2ETHE @ A_27a @ V15x ) ) ) )
& ! [V16x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EIS__SOME @ A_27a @ V16x )
=> ( ( c_2Eoption_2Eoption__CASE @ A_27a @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V16x @ V1e @ ( c_2Eoption_2ESOME @ A_27a ) )
= V16x ) )
& ! [V17v: A_27b,V18f: A_27a > A_27b] :
( ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b @ ( c_2Eoption_2ENONE @ A_27a ) @ V17v @ V18f )
= V17v )
& ! [V19x: A_27a,V20v: A_27b,V21f: A_27a > A_27b] :
( ( c_2Eoption_2Eoption__CASE @ A_27a @ A_27b @ ( c_2Eoption_2ESOME @ A_27a @ V19x ) @ V20v @ V21f )
= ( V21f @ V19x ) )
& ! [V22f: A_27a > A_27b,V23x: A_27a] :
( ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V22f @ ( c_2Eoption_2ESOME @ A_27a @ V23x ) )
= ( c_2Eoption_2ESOME @ A_27b @ ( V22f @ V23x ) ) )
& ! [V24f: A_27a > A_27b] :
( ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V24f @ ( c_2Eoption_2ENONE @ A_27a ) )
= ( c_2Eoption_2ENONE @ A_27b ) )
& ( ( c_2Eoption_2EOPTION__JOIN @ A_27a @ ( c_2Eoption_2ENONE @ ( tyop_2Eoption_2Eoption @ A_27a ) ) )
= ( c_2Eoption_2ENONE @ A_27a ) )
& ! [V25x: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EOPTION__JOIN @ A_27a @ ( c_2Eoption_2ESOME @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V25x ) )
= V25x ) ) ).
thf(thm_2Eoption_2EOPTREL__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o,V1x: tyop_2Eoption_2Eoption @ A_27a,V2y: tyop_2Eoption_2Eoption @ A_27b] :
( ( c_2Eoption_2EOPTREL @ A_27a @ A_27b @ V0R @ V1x @ V2y )
<=> ( ( ( V1x
= ( c_2Eoption_2ENONE @ A_27a ) )
& ( V2y
= ( c_2Eoption_2ENONE @ A_27b ) ) )
| ? [V3x0: A_27a,V4y0: A_27b] :
( ( V1x
= ( c_2Eoption_2ESOME @ A_27a @ V3x0 ) )
& ( V2y
= ( c_2Eoption_2ESOME @ A_27b @ V4y0 ) )
& ( V0R @ V3x0 @ V4y0 ) ) ) ) ).
thf(thm_2Equotient__option_2EOPTION__REL__def,conjecture,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2R: A_27a > A_27a > $o] :
( ( ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V2R @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ENONE @ A_27a ) )
= c_2Ebool_2ET )
& ( ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V2R @ ( c_2Eoption_2ESOME @ A_27a @ V1x ) @ ( c_2Eoption_2ENONE @ A_27a ) )
= c_2Ebool_2EF )
& ( ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V2R @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ESOME @ A_27a @ V0y ) )
= c_2Ebool_2EF )
& ( ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V2R @ ( c_2Eoption_2ESOME @ A_27a @ V1x ) @ ( c_2Eoption_2ESOME @ A_27a @ V0y ) )
= ( V2R @ V1x @ V0y ) ) ) ).
%------------------------------------------------------------------------------