ITP001 Axioms: ITP071^7.ax
%------------------------------------------------------------------------------
% File : ITP071^7 : 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 : quotient_option.ax [Gau19]
% : HL4071^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 45 ( 7 unt; 23 typ; 0 def)
% Number of atoms : 65 ( 14 equ; 1 cnn)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 316 ( 1 ~; 1 |; 5 &; 283 @)
% ( 7 <=>; 19 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 10 avg; 283 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 116 ( 116 >; 0 *; 0 +; 0 <<)
% Number of symbols : 24 ( 22 usr; 3 con; 0-8 aty)
% Number of variables : 125 ( 0 ^ 103 !; 1 ?; 125 :)
% ( 21 !>; 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_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_2Equotient_2E_2D_2D_3E,type,
c_2Equotient_2E_2D_2D_3E:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( A_27c > A_27b ) > A_27a > A_27d ) ).
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_2Equotient_2E_3D_3D_3D_3E,type,
c_2Equotient_2E_3D_3D_3D_3E:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( A_27a > A_27b ) > ( A_27a > A_27b ) > $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_2Equotient_2EEQUIV,type,
c_2Equotient_2EEQUIV:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ebool_2EF,type,
c_2Ebool_2EF: $o ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
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__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_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > A_27a ) > $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_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_2Equotient__option_2EOPTION__MAP__I,axiom,
! [A_27a: $tType] :
( ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27a @ ( c_2Ecombin_2EI @ A_27a ) )
= ( c_2Ecombin_2EI @ ( tyop_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(thm_2Equotient__option_2EOPTION__REL__def,axiom,
! [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 ) ) ) ).
thf(thm_2Equotient__option_2EOPTION__REL__EQ,axiom,
! [A_27a: $tType] :
( ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ ( c_2Emin_2E_3D @ A_27a ) )
= ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(thm_2Equotient__option_2EOPTION__EQUIV,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V0R )
=> ( c_2Equotient_2EEQUIV @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V0R ) ) ) ).
thf(thm_2Equotient__option_2EOPTION__QUOTIENT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( c_2Equotient_2EQUOTIENT @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27b ) @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V0R ) @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V1abs ) @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a @ V2rep ) ) ) ).
thf(thm_2Equotient__option_2ENONE__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( ( c_2Eoption_2ENONE @ A_27b )
= ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V1abs @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ).
thf(thm_2Equotient__option_2ENONE__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V0R @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ).
thf(thm_2Equotient__option_2ESOME__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: A_27b] :
( ( c_2Eoption_2ESOME @ A_27b @ V3x )
= ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V1abs @ ( c_2Eoption_2ESOME @ A_27a @ ( V2rep @ V3x ) ) ) ) ) ).
thf(thm_2Equotient__option_2ESOME__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: A_27a,V4y: A_27a] :
( ( V0R @ V3x @ V4y )
=> ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V0R @ ( c_2Eoption_2ESOME @ A_27a @ V3x ) @ ( c_2Eoption_2ESOME @ A_27a @ V4y ) ) ) ) ).
thf(thm_2Equotient__option_2EIS__SOME__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: tyop_2Eoption_2Eoption @ A_27b] :
( ( c_2Eoption_2EIS__SOME @ A_27b @ V3x )
= ( c_2Eoption_2EIS__SOME @ A_27a @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a @ V2rep @ V3x ) ) ) ) ).
thf(thm_2Equotient__option_2EIS__SOME__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: tyop_2Eoption_2Eoption @ A_27a,V4y: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V0R @ V3x @ V4y )
=> ( ( c_2Eoption_2EIS__SOME @ A_27a @ V3x )
= ( c_2Eoption_2EIS__SOME @ A_27a @ V4y ) ) ) ) ).
thf(thm_2Equotient__option_2EIS__NONE__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: tyop_2Eoption_2Eoption @ A_27b] :
( ( c_2Eoption_2EIS__NONE @ A_27b @ V3x )
= ( c_2Eoption_2EIS__NONE @ A_27a @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a @ V2rep @ V3x ) ) ) ) ).
thf(thm_2Equotient__option_2EIS__NONE__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: tyop_2Eoption_2Eoption @ A_27a,V4y: tyop_2Eoption_2Eoption @ A_27a] :
( ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V0R @ V3x @ V4y )
=> ( ( c_2Eoption_2EIS__NONE @ A_27a @ V3x )
= ( c_2Eoption_2EIS__NONE @ A_27a @ V4y ) ) ) ) ).
thf(thm_2Equotient__option_2EOPTION__MAP__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6a: tyop_2Eoption_2Eoption @ A_27c,V7f: A_27c > A_27d] :
( ( c_2Eoption_2EOPTION__MAP @ A_27c @ A_27d @ V7f @ V6a )
= ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27d @ V4abs2 @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ V7f ) @ ( c_2Eoption_2EOPTION__MAP @ A_27c @ A_27a @ V2rep1 @ V6a ) ) ) ) ) ) ).
thf(thm_2Equotient__option_2EOPTION__MAP__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6a1: tyop_2Eoption_2Eoption @ A_27a,V7a2: tyop_2Eoption_2Eoption @ A_27a,V8f1: A_27a > A_27b,V9f2: A_27a > A_27b] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V8f1 @ V9f2 )
& ( c_2Eoption_2EOPTREL @ A_27a @ A_27a @ V0R1 @ V6a1 @ V7a2 ) )
=> ( c_2Eoption_2EOPTREL @ A_27b @ A_27b @ V3R2 @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V8f1 @ V6a1 ) @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b @ V9f2 @ V7a2 ) ) ) ) ) ).
%------------------------------------------------------------------------------