TPTP Problem File: ITP002_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP002_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory 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_5.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 1240 ( 254 unt; 301 typ; 0 def)
% Number of atoms : 9373 ( 498 equ)
% Maximal formula atoms : 25 ( 7 avg)
% Number of connectives : 3069 ( 141 ~; 151 |; 479 &)
% ( 357 <=>;1941 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of FOOLs : 5506 (5506 fml; 0 var)
% Number of types : 8 ( 6 usr)
% Number of type conns : 553 ( 259 >; 294 *; 0 +; 0 <<)
% Number of predicates : 19 ( 16 usr; 7 prp; 0-2 aty)
% Number of functors : 293 ( 293 usr; 36 con; 0-5 aty)
% Number of variables : 3073 (2979 !; 94 ?;3073 :)
% SPC : TF0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001_2.ax').
include('Axioms/ITP001/ITP002_5.ax').
include('Axioms/ITP001/ITP003_5.ax').
include('Axioms/ITP001/ITP004_5.ax').
include('Axioms/ITP001/ITP007_5.ax').
include('Axioms/ITP001/ITP006_5.ax').
include('Axioms/ITP001/ITP005_5.ax').
include('Axioms/ITP001/ITP008_5.ax').
include('Axioms/ITP001/ITP009_5.ax').
include('Axioms/ITP001/ITP010_5.ax').
include('Axioms/ITP001/ITP012_5.ax').
include('Axioms/ITP001/ITP011_5.ax').
%------------------------------------------------------------------------------
tff(tp_ty_2Eoption_2Eoption,type,
ty_2Eoption_2Eoption: del > del ).
tff(tp_c_2Eoption_2EIS__NONE,type,
c_2Eoption_2EIS__NONE: del > $i ).
tff(mem_c_2Eoption_2EIS__NONE,axiom,
! [A_27a: del] : mem(c_2Eoption_2EIS__NONE(A_27a),arr(ty_2Eoption_2Eoption(A_27a),bool)) ).
tff(tp_c_2Eoption_2EIS__SOME,type,
c_2Eoption_2EIS__SOME: del > $i ).
tff(mem_c_2Eoption_2EIS__SOME,axiom,
! [A_27a: del] : mem(c_2Eoption_2EIS__SOME(A_27a),arr(ty_2Eoption_2Eoption(A_27a),bool)) ).
tff(tp_c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE: del > $i ).
tff(mem_c_2Eoption_2ENONE,axiom,
! [A_27a: del] : mem(c_2Eoption_2ENONE(A_27a),ty_2Eoption_2Eoption(A_27a)) ).
tff(tp_c_2Eoption_2EOPTION__ALL,type,
c_2Eoption_2EOPTION__ALL: del > $i ).
tff(mem_c_2Eoption_2EOPTION__ALL,axiom,
! [A_27a: del] : mem(c_2Eoption_2EOPTION__ALL(A_27a),arr(arr(A_27a,bool),arr(ty_2Eoption_2Eoption(A_27a),bool))) ).
tff(tp_c_2Eoption_2EOPTION__APPLY,type,
c_2Eoption_2EOPTION__APPLY: ( del * del ) > $i ).
tff(mem_c_2Eoption_2EOPTION__APPLY,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eoption_2EOPTION__APPLY(A_27a,A_27b),arr(ty_2Eoption_2Eoption(arr(A_27b,A_27a)),arr(ty_2Eoption_2Eoption(A_27b),ty_2Eoption_2Eoption(A_27a)))) ).
tff(tp_c_2Eoption_2EOPTION__BIND,type,
c_2Eoption_2EOPTION__BIND: ( del * del ) > $i ).
tff(mem_c_2Eoption_2EOPTION__BIND,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eoption_2EOPTION__BIND(A_27a,A_27b),arr(ty_2Eoption_2Eoption(A_27b),arr(arr(A_27b,ty_2Eoption_2Eoption(A_27a)),ty_2Eoption_2Eoption(A_27a)))) ).
tff(tp_c_2Eoption_2EOPTION__CHOICE,type,
c_2Eoption_2EOPTION__CHOICE: del > $i ).
tff(mem_c_2Eoption_2EOPTION__CHOICE,axiom,
! [A_27a: del] : mem(c_2Eoption_2EOPTION__CHOICE(A_27a),arr(ty_2Eoption_2Eoption(A_27a),arr(ty_2Eoption_2Eoption(A_27a),ty_2Eoption_2Eoption(A_27a)))) ).
tff(stp_c_ty_2Eoption_2Eoption_ty_2Eone_2Eone,type,
tp__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone: $tType ).
tff(stp_inj_c_ty_2Eoption_2Eoption_ty_2Eone_2Eone,type,
inj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone: tp__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone > $i ).
tff(stp_surj_c_ty_2Eoption_2Eoption_ty_2Eone_2Eone,type,
surj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone: $i > tp__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone ).
tff(stp_inj_surj_c_ty_2Eoption_2Eoption_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone] : surj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone(inj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone(X)) = X ).
tff(stp_inj_mem_c_ty_2Eoption_2Eoption_ty_2Eone_2Eone,axiom,
! [X: tp__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone] : mem(inj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone(X),ty_2Eoption_2Eoption(ty_2Eone_2Eone)) ).
tff(stp_iso_mem_c_ty_2Eoption_2Eoption_ty_2Eone_2Eone,axiom,
! [X: $i] :
( mem(X,ty_2Eoption_2Eoption(ty_2Eone_2Eone))
=> ( X = inj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone(surj__c_ty_2Eoption_2Eoption_ty_2Eone_2Eone(X)) ) ) ).
tff(tp_c_2Eoption_2EOPTION__GUARD,type,
c_2Eoption_2EOPTION__GUARD: $i ).
tff(mem_c_2Eoption_2EOPTION__GUARD,axiom,
mem(c_2Eoption_2EOPTION__GUARD,arr(bool,ty_2Eoption_2Eoption(ty_2Eone_2Eone))) ).
tff(tp_c_2Eoption_2EOPTION__IGNORE__BIND,type,
c_2Eoption_2EOPTION__IGNORE__BIND: ( del * del ) > $i ).
tff(mem_c_2Eoption_2EOPTION__IGNORE__BIND,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eoption_2EOPTION__IGNORE__BIND(A_27a,A_27b),arr(ty_2Eoption_2Eoption(A_27b),arr(ty_2Eoption_2Eoption(A_27a),ty_2Eoption_2Eoption(A_27a)))) ).
tff(tp_c_2Eoption_2EOPTION__JOIN,type,
c_2Eoption_2EOPTION__JOIN: del > $i ).
tff(mem_c_2Eoption_2EOPTION__JOIN,axiom,
! [A_27a: del] : mem(c_2Eoption_2EOPTION__JOIN(A_27a),arr(ty_2Eoption_2Eoption(ty_2Eoption_2Eoption(A_27a)),ty_2Eoption_2Eoption(A_27a))) ).
tff(tp_c_2Eoption_2EOPTION__MAP,type,
c_2Eoption_2EOPTION__MAP: ( del * del ) > $i ).
tff(mem_c_2Eoption_2EOPTION__MAP,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(ty_2Eoption_2Eoption(A_27a),ty_2Eoption_2Eoption(A_27b)))) ).
tff(tp_c_2Eoption_2EOPTION__MAP2,type,
c_2Eoption_2EOPTION__MAP2: ( del * del * del ) > $i ).
tff(mem_c_2Eoption_2EOPTION__MAP2,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Eoption_2EOPTION__MAP2(A_27a,A_27b,A_27c),arr(arr(A_27b,arr(A_27c,A_27a)),arr(ty_2Eoption_2Eoption(A_27b),arr(ty_2Eoption_2Eoption(A_27c),ty_2Eoption_2Eoption(A_27a))))) ).
tff(tp_c_2Eoption_2EOPTION__MCOMP,type,
c_2Eoption_2EOPTION__MCOMP: ( del * del * del ) > $i ).
tff(mem_c_2Eoption_2EOPTION__MCOMP,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Eoption_2EOPTION__MCOMP(A_27a,A_27b,A_27c),arr(arr(A_27b,ty_2Eoption_2Eoption(A_27a)),arr(arr(A_27c,ty_2Eoption_2Eoption(A_27b)),arr(A_27c,ty_2Eoption_2Eoption(A_27a))))) ).
tff(tp_c_2Eoption_2EOPTREL,type,
c_2Eoption_2EOPTREL: ( del * del ) > $i ).
tff(mem_c_2Eoption_2EOPTREL,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eoption_2EOPTREL(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),arr(ty_2Eoption_2Eoption(A_27a),arr(ty_2Eoption_2Eoption(A_27b),bool)))) ).
tff(tp_c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME: del > $i ).
tff(mem_c_2Eoption_2ESOME,axiom,
! [A_27a: del] : mem(c_2Eoption_2ESOME(A_27a),arr(A_27a,ty_2Eoption_2Eoption(A_27a))) ).
tff(tp_c_2Eoption_2ETHE,type,
c_2Eoption_2ETHE: del > $i ).
tff(mem_c_2Eoption_2ETHE,axiom,
! [A_27a: del] : mem(c_2Eoption_2ETHE(A_27a),arr(ty_2Eoption_2Eoption(A_27a),A_27a)) ).
tff(tp_c_2Eoption_2Eoption__ABS,type,
c_2Eoption_2Eoption__ABS: del > $i ).
tff(mem_c_2Eoption_2Eoption__ABS,axiom,
! [A_27a: del] : mem(c_2Eoption_2Eoption__ABS(A_27a),arr(ty_2Esum_2Esum(A_27a,ty_2Eone_2Eone),ty_2Eoption_2Eoption(A_27a))) ).
tff(tp_c_2Eoption_2Eoption__CASE,type,
c_2Eoption_2Eoption__CASE: ( del * del ) > $i ).
tff(mem_c_2Eoption_2Eoption__CASE,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eoption_2Eoption__CASE(A_27a,A_27b),arr(ty_2Eoption_2Eoption(A_27a),arr(A_27b,arr(arr(A_27a,A_27b),A_27b)))) ).
tff(tp_c_2Eoption_2Eoption__REP,type,
c_2Eoption_2Eoption__REP: del > $i ).
tff(mem_c_2Eoption_2Eoption__REP,axiom,
! [A_27a: del] : mem(c_2Eoption_2Eoption__REP(A_27a),arr(ty_2Eoption_2Eoption(A_27a),ty_2Esum_2Esum(A_27a,ty_2Eone_2Eone))) ).
tff(tp_c_2Eoption_2Esome,type,
c_2Eoption_2Esome: del > $i ).
tff(mem_c_2Eoption_2Esome,axiom,
! [A_27a: del] : mem(c_2Eoption_2Esome(A_27a),arr(arr(A_27a,bool),ty_2Eoption_2Eoption(A_27a))) ).
tff(ax_thm_2Eoption_2Eoption__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( mem(V0rep,arr(ty_2Eoption_2Eoption(A_27a),ty_2Esum_2Esum(A_27a,ty_2Eone_2Eone)))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(ty_2Esum_2Esum(A_27a,ty_2Eone_2Eone),ty_2Eoption_2Eoption(A_27a)),k(ty_2Esum_2Esum(A_27a,ty_2Eone_2Eone),c_2Ebool_2ET)),V0rep)) ) ).
tff(ax_thm_2Eoption_2Eoption__REP__ABS__DEF,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( mem(V0a,ty_2Eoption_2Eoption(A_27a))
=> ( ap(c_2Eoption_2Eoption__ABS(A_27a),ap(c_2Eoption_2Eoption__REP(A_27a),V0a)) = V0a ) )
& ! [V1r: $i] :
( mem(V1r,ty_2Esum_2Esum(A_27a,ty_2Eone_2Eone))
=> ( p(ap(k(ty_2Esum_2Esum(A_27a,ty_2Eone_2Eone),c_2Ebool_2ET),V1r))
<=> ( ap(c_2Eoption_2Eoption__REP(A_27a),ap(c_2Eoption_2Eoption__ABS(A_27a),V1r)) = V1r ) ) ) ) ).
tff(ax_thm_2Eoption_2ESOME__DEF,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ap(c_2Eoption_2ESOME(A_27a),V0x) = ap(c_2Eoption_2Eoption__ABS(A_27a),ap(c_2Esum_2EINL(A_27a,ty_2Eone_2Eone),V0x)) ) ) ).
tff(ax_thm_2Eoption_2ENONE__DEF,axiom,
! [A_27a: del] : c_2Eoption_2ENONE(A_27a) = ap(c_2Eoption_2Eoption__ABS(A_27a),ap(c_2Esum_2EINR(A_27a,ty_2Eone_2Eone),inj__ty_2Eone_2Eone(fo__c_2Eone_2Eone))) ).
tff(conj_thm_2Eoption_2Eoption__Axiom,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( mem(V0e,A_27b)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ? [V2fn: $i] :
( mem(V2fn,arr(ty_2Eoption_2Eoption(A_27a),A_27b))
& ( ap(V2fn,c_2Eoption_2ENONE(A_27a)) = V0e )
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ap(V2fn,ap(c_2Eoption_2ESOME(A_27a),V3x)) = ap(V1f,V3x) ) ) ) ) ) ).
tff(conj_thm_2Eoption_2Eoption__induction,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Eoption_2Eoption(A_27a),bool))
=> ( ( p(ap(V0P,c_2Eoption_2ENONE(A_27a)))
& ! [V1a: $i] :
( mem(V1a,A_27a)
=> p(ap(V0P,ap(c_2Eoption_2ESOME(A_27a),V1a))) ) )
=> ! [V2x: $i] :
( mem(V2x,ty_2Eoption_2Eoption(A_27a))
=> p(ap(V0P,V2x)) ) ) ) ).
tff(conj_thm_2Eoption_2Eoption__nchotomy,axiom,
! [A_27a: del,V0opt: $i] :
( mem(V0opt,ty_2Eoption_2Eoption(A_27a))
=> ( ( V0opt = c_2Eoption_2ENONE(A_27a) )
| ? [V1x: $i] :
( mem(V1x,A_27a)
& ( V0opt = ap(c_2Eoption_2ESOME(A_27a),V1x) ) ) ) ) ).
tff(ax_thm_2Eoption_2Eoption__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0v: $i] :
( mem(V0v,A_27b)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),c_2Eoption_2ENONE(A_27a)),V0v),V1f) = V0v ) ) )
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3v: $i] :
( mem(V3v,A_27b)
=> ! [V4f: $i] :
( mem(V4f,arr(A_27a,A_27b))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),ap(c_2Eoption_2ESOME(A_27a),V2x)),V3v),V4f) = ap(V4f,V2x) ) ) ) ) ) ).
tff(conj_thm_2Eoption_2EFORALL__OPTION,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Eoption_2Eoption(A_27a),bool))
=> ( ! [V1opt: $i] :
( mem(V1opt,ty_2Eoption_2Eoption(A_27a))
=> p(ap(V0P,V1opt)) )
<=> ( p(ap(V0P,c_2Eoption_2ENONE(A_27a)))
& ! [V2x: $i] :
( mem(V2x,A_27a)
=> p(ap(V0P,ap(c_2Eoption_2ESOME(A_27a),V2x))) ) ) ) ) ).
tff(conj_thm_2Eoption_2EEXISTS__OPTION,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Eoption_2Eoption(A_27a),bool))
=> ( ? [V1opt: $i] :
( mem(V1opt,ty_2Eoption_2Eoption(A_27a))
& p(ap(V0P,V1opt)) )
<=> ( p(ap(V0P,c_2Eoption_2ENONE(A_27a)))
| ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(V0P,ap(c_2Eoption_2ESOME(A_27a),V2x))) ) ) ) ) ).
tff(conj_thm_2Eoption_2ESOME__11,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27a)
=> ( ( ap(c_2Eoption_2ESOME(A_27a),V0x) = ap(c_2Eoption_2ESOME(A_27a),V1y) )
<=> ( V0x = V1y ) ) ) ) ).
tff(conj_thm_2Eoption_2ENOT__NONE__SOME,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( c_2Eoption_2ENONE(A_27a) != ap(c_2Eoption_2ESOME(A_27a),V0x) ) ) ).
tff(conj_thm_2Eoption_2ENOT__SOME__NONE,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ap(c_2Eoption_2ESOME(A_27a),V0x) != c_2Eoption_2ENONE(A_27a) ) ) ).
tff(ax_thm_2Eoption_2EOPTION__MAP__DEF,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V0f),ap(c_2Eoption_2ESOME(A_27a),V1x)) = ap(c_2Eoption_2ESOME(A_27b),ap(V0f,V1x)) ) ) )
& ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V2f),c_2Eoption_2ENONE(A_27a)) = c_2Eoption_2ENONE(A_27b) ) ) ) ).
tff(ax_thm_2Eoption_2EIS__SOME__DEF,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( mem(V0x,A_27a)
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(c_2Eoption_2ESOME(A_27a),V0x)))
<=> $true ) )
& ( p(ap(c_2Eoption_2EIS__SOME(A_27a),c_2Eoption_2ENONE(A_27a)))
<=> $false ) ) ).
tff(ax_thm_2Eoption_2EIS__NONE__DEF,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( mem(V0x,A_27a)
=> ( p(ap(c_2Eoption_2EIS__NONE(A_27a),ap(c_2Eoption_2ESOME(A_27a),V0x)))
<=> $false ) )
& ( p(ap(c_2Eoption_2EIS__NONE(A_27a),c_2Eoption_2ENONE(A_27a)))
<=> $true ) ) ).
tff(ax_thm_2Eoption_2ETHE__DEF,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ap(c_2Eoption_2ETHE(A_27a),ap(c_2Eoption_2ESOME(A_27a),V0x)) = V0x ) ) ).
tff(ax_thm_2Eoption_2EOPTION__MAP2__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27b,arr(A_27c,A_27a)))
=> ! [V1x: $i] :
( mem(V1x,ty_2Eoption_2Eoption(A_27b))
=> ! [V2y: $i] :
( mem(V2y,ty_2Eoption_2Eoption(A_27c))
=> ( ap(ap(ap(c_2Eoption_2EOPTION__MAP2(A_27a,A_27b,A_27c),V0f),V1x),V2y) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eoption_2Eoption(A_27a)),ap(ap(c_2Ebool_2E_2F_5C,ap(c_2Eoption_2EIS__SOME(A_27b),V1x)),ap(c_2Eoption_2EIS__SOME(A_27c),V2y))),ap(c_2Eoption_2ESOME(A_27a),ap(ap(V0f,ap(c_2Eoption_2ETHE(A_27b),V1x)),ap(c_2Eoption_2ETHE(A_27c),V2y)))),c_2Eoption_2ENONE(A_27a)) ) ) ) ) ).
tff(ax_thm_2Eoption_2EOPTION__JOIN__DEF,axiom,
! [A_27a: del] :
( ( ap(c_2Eoption_2EOPTION__JOIN(A_27a),c_2Eoption_2ENONE(ty_2Eoption_2Eoption(A_27a))) = c_2Eoption_2ENONE(A_27a) )
& ! [V0x: $i] :
( mem(V0x,ty_2Eoption_2Eoption(A_27a))
=> ( ap(c_2Eoption_2EOPTION__JOIN(A_27a),ap(c_2Eoption_2ESOME(ty_2Eoption_2Eoption(A_27a)),V0x)) = V0x ) ) ) ).
tff(conj_thm_2Eoption_2EOPTION__MAP2__THM,conjecture,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27b,arr(A_27c,A_27a)))
=> ! [V1x: $i] :
( mem(V1x,A_27b)
=> ! [V2y: $i] :
( mem(V2y,A_27c)
=> ( ( ap(ap(ap(c_2Eoption_2EOPTION__MAP2(A_27a,A_27b,A_27c),V0f),ap(c_2Eoption_2ESOME(A_27b),V1x)),ap(c_2Eoption_2ESOME(A_27c),V2y)) = ap(c_2Eoption_2ESOME(A_27a),ap(ap(V0f,V1x),V2y)) )
& ( ap(ap(ap(c_2Eoption_2EOPTION__MAP2(A_27a,A_27b,A_27c),V0f),ap(c_2Eoption_2ESOME(A_27b),V1x)),c_2Eoption_2ENONE(A_27c)) = c_2Eoption_2ENONE(A_27a) )
& ( ap(ap(ap(c_2Eoption_2EOPTION__MAP2(A_27a,A_27b,A_27c),V0f),c_2Eoption_2ENONE(A_27b)),ap(c_2Eoption_2ESOME(A_27c),V2y)) = c_2Eoption_2ENONE(A_27a) )
& ( ap(ap(ap(c_2Eoption_2EOPTION__MAP2(A_27a,A_27b,A_27c),V0f),c_2Eoption_2ENONE(A_27b)),c_2Eoption_2ENONE(A_27c)) = c_2Eoption_2ENONE(A_27a) ) ) ) ) ) ).
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