TPTP Problem File: ITP011_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP011_2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory 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_2.p [TPAP]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.80 v8.1.0, 0.82 v7.5.0
% Syntax : Number of formulae : 90 ( 29 unt; 37 typ; 0 def)
% Number of atoms : 400 ( 38 equ)
% Maximal formula atoms : 55 ( 4 avg)
% Number of connectives : 175 ( 11 ~; 7 |; 48 &)
% ( 42 <=>; 67 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 183 ( 183 fml; 0 var)
% Number of types : 4 ( 2 usr)
% Number of type conns : 35 ( 25 >; 10 *; 0 +; 0 <<)
% Number of predicates : 8 ( 5 usr; 3 prp; 0-2 aty)
% Number of functors : 33 ( 33 usr; 10 con; 0-2 aty)
% Number of variables : 108 ( 104 !; 4 ?; 108 :)
% SPC : TF0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001_2.ax').
%------------------------------------------------------------------------------
tff(tp_ty_2Eoption_2Eoption,type,
ty_2Eoption_2Eoption: del > del ).
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__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(stp_o,type,
tp__o: $tType ).
tff(stp_inj_o,type,
inj__o: tp__o > $i ).
tff(stp_surj_o,type,
surj__o: $i > tp__o ).
tff(stp_inj_surj_o,axiom,
! [X: tp__o] : surj__o(inj__o(X)) = X ).
tff(stp_inj_mem_o,axiom,
! [X: tp__o] : mem(inj__o(X),bool) ).
tff(stp_iso_mem_o,axiom,
! [X: $i] :
( mem(X,bool)
=> ( X = inj__o(surj__o(X)) ) ) ).
tff(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
tff(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem(c_2Emin_2E_3D_3D_3E,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Emin_2E_3D_3D_3E,type,
fo__c_2Emin_2E_3D_3D_3E: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Emin_2E_3D_3D_3E,axiom,
! [X0: tp__o,X1: tp__o] : inj__o(fo__c_2Emin_2E_3D_3D_3E(X0,X1)) = ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(X0)),inj__o(X1)) ).
tff(ax_imp_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Emin_2E_3D_3D_3E,Q),R))
<=> ( p(Q)
=> p(R) ) ) ) ) ).
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_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
tff(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
tff(stp_fo_c_2Ebool_2EF,type,
fo__c_2Ebool_2EF: tp__o ).
tff(stp_eq_fo_c_2Ebool_2EF,axiom,
inj__o(fo__c_2Ebool_2EF) = c_2Ebool_2EF ).
tff(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
tff(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
tff(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
tff(stp_fo_c_2Ebool_2ET,type,
fo__c_2Ebool_2ET: tp__o ).
tff(stp_eq_fo_c_2Ebool_2ET,axiom,
inj__o(fo__c_2Ebool_2ET) = c_2Ebool_2ET ).
tff(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
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_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
tff(mem_c_2Ebool_2E_7E,axiom,
mem(c_2Ebool_2E_7E,arr(bool,bool)) ).
tff(stp_fo_c_2Ebool_2E_7E,type,
fo__c_2Ebool_2E_7E: tp__o > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_7E,axiom,
! [X0: tp__o] : inj__o(fo__c_2Ebool_2E_7E(X0)) = ap(c_2Ebool_2E_7E,inj__o(X0)) ).
tff(ax_neg_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ( p(ap(c_2Ebool_2E_7E,Q))
<=> ~ p(Q) ) ) ).
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_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_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
tff(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : mem(c_2Ebool_2E_3F(A_27a),arr(arr(A_27a,bool),bool)) ).
tff(ax_ex_p,axiom,
! [A: del,Q: $i] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_3F(A),Q))
<=> ? [X: $i] :
( mem(X,A)
& p(ap(Q,X)) ) ) ) ).
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_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
tff(mem_c_2Ebool_2E_2F_5C,axiom,
mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Ebool_2E_2F_5C,type,
fo__c_2Ebool_2E_2F_5C: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_2F_5C,axiom,
! [X0: tp__o,X1: tp__o] : inj__o(fo__c_2Ebool_2E_2F_5C(X0,X1)) = ap(ap(c_2Ebool_2E_2F_5C,inj__o(X0)),inj__o(X1)) ).
tff(ax_and_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_2F_5C,Q),R))
<=> ( p(Q)
& p(R) ) ) ) ) ).
tff(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
tff(mem_c_2Ebool_2E_5C_2F,axiom,
mem(c_2Ebool_2E_5C_2F,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Ebool_2E_5C_2F,type,
fo__c_2Ebool_2E_5C_2F: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_5C_2F,axiom,
! [X0: tp__o,X1: tp__o] : inj__o(fo__c_2Ebool_2E_5C_2F(X0,X1)) = ap(ap(c_2Ebool_2E_5C_2F,inj__o(X0)),inj__o(X1)) ).
tff(ax_or_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_5C_2F,Q),R))
<=> ( p(Q)
| p(R) ) ) ) ) ).
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_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
tff(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : mem(c_2Emin_2E_3D(A_27a),arr(A_27a,arr(A_27a,bool))) ).
tff(ax_eq_p,axiom,
! [A: del,X: $i] :
( mem(X,A)
=> ! [Y: $i] :
( mem(Y,A)
=> ( p(ap(ap(c_2Emin_2E_3D(A),X),Y))
<=> ( X = Y ) ) ) ) ).
tff(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
tff(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ).
tff(ax_all_p,axiom,
! [A: del,Q: $i] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_21(A),Q))
<=> ! [X: $i] :
( mem(X,A)
=> p(ap(Q,X)) ) ) ) ).
tff(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
tff(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
! [V0t1: tp__o,V1t2: tp__o] :
( ( p(inj__o(V0t1))
=> p(inj__o(V1t2)) )
=> ( ( p(inj__o(V1t2))
=> p(inj__o(V0t1)) )
=> ( p(inj__o(V0t1))
<=> p(inj__o(V1t2)) ) ) ) ).
tff(conj_thm_2Ebool_2EFALSITY,axiom,
! [V0t: tp__o] :
( $false
=> p(inj__o(V0t)) ) ).
tff(conj_thm_2Ebool_2EEXISTS__SIMP,axiom,
! [A_27a: del,V0t: tp__o] :
( ? [V1x: $i] :
( mem(V1x,A_27a)
& p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
& p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
& $true )
<=> p(inj__o(V0t)) )
& ( ( $false
& p(inj__o(V0t)) )
<=> $false )
& ( ( p(inj__o(V0t))
& $false )
<=> $false )
& ( ( p(inj__o(V0t))
& p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2EOR__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
| p(inj__o(V0t)) )
<=> $true )
& ( ( p(inj__o(V0t))
| $true )
<=> $true )
& ( ( $false
| p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
| $false )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
| p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: tp__o] :
( ~ ~ p(inj__o(V0t))
<=> p(inj__o(V0t)) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
tff(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ( V0x = V0x )
<=> $true ) ) ).
tff(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
<=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
<=> $true )
<=> p(inj__o(V0t)) )
& ( ( $false
<=> p(inj__o(V0t)) )
<=> ~ p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
<=> $false )
<=> ~ p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Eoption_2Eoption__CLAUSES,axiom,
! [A_27a: del,A_27b: del,V0e: $i] :
( mem(V0e,A_27b)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27b))
=> ! [V2e: $i] :
( mem(V2e,ty_2Eoption_2Eoption(A_27a))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ! [V4y: $i] :
( mem(V4y,A_27a)
=> ( ( ap(c_2Eoption_2ESOME(A_27a),V3x) = ap(c_2Eoption_2ESOME(A_27a),V4y) )
<=> ( V3x = V4y ) ) ) )
& ! [V5x: $i] :
( mem(V5x,A_27a)
=> ( ap(c_2Eoption_2ETHE(A_27a),ap(c_2Eoption_2ESOME(A_27a),V5x)) = V5x ) )
& ! [V6x: $i] :
( mem(V6x,A_27a)
=> ( c_2Eoption_2ENONE(A_27a) != ap(c_2Eoption_2ESOME(A_27a),V6x) ) )
& ! [V7x: $i] :
( mem(V7x,A_27a)
=> ( ap(c_2Eoption_2ESOME(A_27a),V7x) != c_2Eoption_2ENONE(A_27a) ) )
& ! [V8x: $i] :
( mem(V8x,A_27a)
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),ap(c_2Eoption_2ESOME(A_27a),V8x)))
<=> $true ) )
& ( p(ap(c_2Eoption_2EIS__SOME(A_27a),c_2Eoption_2ENONE(A_27a)))
<=> $false )
& ! [V9x: $i] :
( mem(V9x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__NONE(A_27a),V9x))
<=> ( V9x = c_2Eoption_2ENONE(A_27a) ) ) )
& ! [V10x: $i] :
( mem(V10x,ty_2Eoption_2Eoption(A_27a))
=> ( ~ p(ap(c_2Eoption_2EIS__SOME(A_27a),V10x))
<=> ( V10x = c_2Eoption_2ENONE(A_27a) ) ) )
& ! [V11x: $i] :
( mem(V11x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),V11x))
=> ( ap(c_2Eoption_2ESOME(A_27a),ap(c_2Eoption_2ETHE(A_27a),V11x)) = V11x ) ) )
& ! [V12x: $i] :
( mem(V12x,ty_2Eoption_2Eoption(A_27a))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,ty_2Eoption_2Eoption(A_27a)),V12x),c_2Eoption_2ENONE(A_27a)),c_2Eoption_2ESOME(A_27a)) = V12x ) )
& ! [V13x: $i] :
( mem(V13x,ty_2Eoption_2Eoption(A_27a))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,ty_2Eoption_2Eoption(A_27a)),V13x),V13x),c_2Eoption_2ESOME(A_27a)) = V13x ) )
& ! [V14x: $i] :
( mem(V14x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__NONE(A_27a),V14x))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),V14x),V0e),V1f) = V0e ) ) )
& ! [V15x: $i] :
( mem(V15x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),V15x))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),V15x),V0e),V1f) = ap(V1f,ap(c_2Eoption_2ETHE(A_27a),V15x)) ) ) )
& ! [V16x: $i] :
( mem(V16x,ty_2Eoption_2Eoption(A_27a))
=> ( p(ap(c_2Eoption_2EIS__SOME(A_27a),V16x))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,ty_2Eoption_2Eoption(A_27a)),V16x),V2e),c_2Eoption_2ESOME(A_27a)) = V16x ) ) )
& ! [V17v: $i] :
( mem(V17v,A_27b)
=> ! [V18f: $i] :
( mem(V18f,arr(A_27a,A_27b))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),c_2Eoption_2ENONE(A_27a)),V17v),V18f) = V17v ) ) )
& ! [V19x: $i] :
( mem(V19x,A_27a)
=> ! [V20v: $i] :
( mem(V20v,A_27b)
=> ! [V21f: $i] :
( mem(V21f,arr(A_27a,A_27b))
=> ( ap(ap(ap(c_2Eoption_2Eoption__CASE(A_27a,A_27b),ap(c_2Eoption_2ESOME(A_27a),V19x)),V20v),V21f) = ap(V21f,V19x) ) ) ) )
& ! [V22f: $i] :
( mem(V22f,arr(A_27a,A_27b))
=> ! [V23x: $i] :
( mem(V23x,A_27a)
=> ( ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V22f),ap(c_2Eoption_2ESOME(A_27a),V23x)) = ap(c_2Eoption_2ESOME(A_27b),ap(V22f,V23x)) ) ) )
& ! [V24f: $i] :
( mem(V24f,arr(A_27a,A_27b))
=> ( ap(ap(c_2Eoption_2EOPTION__MAP(A_27a,A_27b),V24f),c_2Eoption_2ENONE(A_27a)) = c_2Eoption_2ENONE(A_27b) ) )
& ( ap(c_2Eoption_2EOPTION__JOIN(A_27a),c_2Eoption_2ENONE(ty_2Eoption_2Eoption(A_27a))) = c_2Eoption_2ENONE(A_27a) )
& ! [V25x: $i] :
( mem(V25x,ty_2Eoption_2Eoption(A_27a))
=> ( ap(c_2Eoption_2EOPTION__JOIN(A_27a),ap(c_2Eoption_2ESOME(ty_2Eoption_2Eoption(A_27a)),V25x)) = V25x ) ) ) ) ) ) ).
tff(ax_thm_2Eoption_2EOPTREL__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ! [V1x: $i] :
( mem(V1x,ty_2Eoption_2Eoption(A_27a))
=> ! [V2y: $i] :
( mem(V2y,ty_2Eoption_2Eoption(A_27b))
=> ( p(ap(ap(ap(c_2Eoption_2EOPTREL(A_27a,A_27b),V0R),V1x),V2y))
<=> ( ( ( V1x = c_2Eoption_2ENONE(A_27a) )
& ( V2y = c_2Eoption_2ENONE(A_27b) ) )
| ? [V3x0: $i] :
( mem(V3x0,A_27a)
& ? [V4y0: $i] :
( mem(V4y0,A_27b)
& ( V1x = ap(c_2Eoption_2ESOME(A_27a),V3x0) )
& ( V2y = ap(c_2Eoption_2ESOME(A_27b),V4y0) )
& p(ap(ap(V0R,V3x0),V4y0)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__option_2EOPTION__REL__def,conjecture,
! [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)) ) ) ) ) ) ).
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