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)
% Maximal formula atoms : 15 ( 26 avg)
% Number of connectives : 88 ( 0 ~; 0 |; 4 &)
% ( 8 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 13 avg)
% Maximal term depth : 1 ( 1 avg)
% Number of FOOLs : 288 ( 288 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 22 ( 19 usr; 4 prp; 0-4 aty)
% Number of functors : 0 ( 0 usr; 0 con; --- aty)
% Number of variables : 88 ( 88 !; 0 ?; 88 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(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)) ) ).
tff(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)) ) ) ) ) ) ).
tff(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)) ) ).
tff(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))) ) ) ).
tff(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))) ) ) ) ) ).
tff(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)) ) ) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------