ITP001 Axioms: ITP071^5.ax
%------------------------------------------------------------------------------
% File : ITP071^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, 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 : quotient_option^2.ax [Gau20]
% : HL4071^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 15 ( 2 unt; 0 typ; 0 def)
% Number of atoms : 391 ( 5 equ; 0 cnn)
% Maximal formula atoms : 56 ( 26 avg)
% Number of connectives : 745 ( 0 ~; 0 |; 4 &; 657 @)
% ( 8 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 20 avg; 657 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 22 ( 19 usr; 21 con; 0-2 aty)
% Number of variables : 88 ( 0 ^ 88 !; 0 ?; 88 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(conj_thm_2Equotient__option_2EOPTION__MAP__I,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) )
= ( c_2Ecombin_2EI @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(conj_thm_2Equotient__option_2EOPTION__REL__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) )
<=> $true )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2y ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V1x ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2y ) ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2EOPTION__REL__EQ,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a ) )
= ( c_2Emin_2E_3D @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ).
thf(conj_thm_2Equotient__option_2EOPTION__EQUIV,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2EOPTION__QUOTIENT,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ( ty_2Eoption_2Eoption @ A_27a ) @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) ) @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V1abs ) ) @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a ) @ V2rep ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2ENONE__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ( ( c_2Eoption_2ENONE @ A_27b )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V1abs ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2ENONE__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2ESOME__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ( ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3x )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V1abs ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ ( ap @ V2rep @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2ESOME__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3x ) ) @ ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2EIS__SOME__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( ty_2Eoption_2Eoption @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27b ) @ V3x ) )
<=> ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27a ) @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a ) @ V2rep ) @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2EIS__SOME__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V4y: $i] :
( ( mem @ V4y @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) @ V3x ) @ V4y ) )
=> ( ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27a ) @ V3x ) )
<=> ( p @ ( ap @ ( c_2Eoption_2EIS__SOME @ A_27a ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2EIS__NONE__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( ty_2Eoption_2Eoption @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Eoption_2EIS__NONE @ A_27b ) @ V3x ) )
<=> ( p @ ( ap @ ( c_2Eoption_2EIS__NONE @ A_27a ) @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27a ) @ V2rep ) @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2EIS__NONE__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V4y: $i] :
( ( mem @ V4y @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R ) @ V3x ) @ V4y ) )
=> ( ( p @ ( ap @ ( c_2Eoption_2EIS__NONE @ A_27a ) @ V3x ) )
<=> ( p @ ( ap @ ( c_2Eoption_2EIS__NONE @ A_27a ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2EOPTION__MAP__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6a: $i] :
( ( mem @ V6a @ ( ty_2Eoption_2Eoption @ A_27c ) )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27c @ A_27d ) )
=> ( ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27c @ A_27d ) @ V7f ) @ V6a )
= ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27b @ A_27d ) @ V4abs2 ) @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V7f ) ) @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27c @ A_27a ) @ V2rep1 ) @ V6a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__option_2EOPTION__MAP__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6a1: $i] :
( ( mem @ V6a1 @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V7a2: $i] :
( ( mem @ V7a2 @ ( ty_2Eoption_2Eoption @ A_27a ) )
=> ! [V8f1: $i] :
( ( mem @ V8f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V9f2: $i] :
( ( mem @ V9f2 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V8f1 ) @ V9f2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27a @ A_27a ) @ V0R1 ) @ V6a1 ) @ V7a2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eoption_2EOPTREL @ A_27b @ A_27b ) @ V3R2 ) @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V8f1 ) @ V6a1 ) ) @ ( ap @ ( ap @ ( c_2Eoption_2EOPTION__MAP @ A_27a @ A_27b ) @ V9f2 ) @ V7a2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------