## TPTP Problem File: ITP006^5.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : ITP006^5 : TPTP v8.1.2. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : HOL4 set theory export of thm_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT.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_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT.p [Gau20]
%          : HL402501^5.p [TPAP]

% Status   : Theorem
% Rating   : 1.00 v7.5.0
% Syntax   : Number of formulae    : 4394 ( 652 unt; 610 typ;   0 def)
%            Number of atoms       : 22155 (2962 equ;   0 cnn)
%            Maximal formula atoms :  319 (   5 avg)
%            Number of connectives : 127889 ( 461   ~; 294   |;2362   &;115560   @)
%                                         (1272 <=>;7940  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   63 (  12 avg)
%            Number of types       :   12 (  10 usr)
%            Number of type conns  :  702 ( 702   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  628 ( 625 usr; 164 con; 0-5 aty)
%            Number of variables   : 13641 (1033   ^12164   !; 444   ?;13641   :)
% SPC      : TH0_THM_EQU_NAR

% 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').
include('Axioms/ITP001/ITP013^5.ax').
include('Axioms/ITP001/ITP014^5.ax').
include('Axioms/ITP001/ITP015^5.ax').
include('Axioms/ITP001/ITP017^5.ax').
include('Axioms/ITP001/ITP016^5.ax').
include('Axioms/ITP001/ITP019^5.ax').
include('Axioms/ITP001/ITP018^5.ax').
include('Axioms/ITP001/ITP021^5.ax').
include('Axioms/ITP001/ITP022^5.ax').
include('Axioms/ITP001/ITP020^5.ax').
include('Axioms/ITP001/ITP024^5.ax').
include('Axioms/ITP001/ITP023^5.ax').
include('Axioms/ITP001/ITP025^5.ax').
include('Axioms/ITP001/ITP026^5.ax').
include('Axioms/ITP001/ITP027^5.ax').
include('Axioms/ITP001/ITP028^5.ax').
include('Axioms/ITP001/ITP031^5.ax').
include('Axioms/ITP001/ITP029^5.ax').
include('Axioms/ITP001/ITP033^5.ax').
include('Axioms/ITP001/ITP030^5.ax').
include('Axioms/ITP001/ITP032^5.ax').
include('Axioms/ITP001/ITP038^5.ax').
include('Axioms/ITP001/ITP035^5.ax').
include('Axioms/ITP001/ITP034^5.ax').
%------------------------------------------------------------------------------
thf(tp_c_2EquantHeuristics_2EGUESS__EXISTS,type,
c_2EquantHeuristics_2EGUESS__EXISTS: del > del > \$i ).

thf(mem_c_2EquantHeuristics_2EGUESS__EXISTS,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).

thf(tp_c_2EquantHeuristics_2EGUESS__EXISTS__GAP,type,
c_2EquantHeuristics_2EGUESS__EXISTS__GAP: del > del > \$i ).

thf(mem_c_2EquantHeuristics_2EGUESS__EXISTS__GAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).

thf(tp_c_2EquantHeuristics_2EGUESS__EXISTS__POINT,type,
c_2EquantHeuristics_2EGUESS__EXISTS__POINT: del > del > \$i ).

thf(mem_c_2EquantHeuristics_2EGUESS__EXISTS__POINT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).

thf(tp_c_2EquantHeuristics_2EGUESS__FORALL,type,
c_2EquantHeuristics_2EGUESS__FORALL: del > del > \$i ).

thf(mem_c_2EquantHeuristics_2EGUESS__FORALL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).

thf(tp_c_2EquantHeuristics_2EGUESS__FORALL__GAP,type,
c_2EquantHeuristics_2EGUESS__FORALL__GAP: del > del > \$i ).

thf(mem_c_2EquantHeuristics_2EGUESS__FORALL__GAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).

thf(tp_c_2EquantHeuristics_2EGUESS__FORALL__POINT,type,
c_2EquantHeuristics_2EGUESS__FORALL__POINT: del > del > \$i ).

thf(mem_c_2EquantHeuristics_2EGUESS__FORALL__POINT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) ).

thf(tp_c_2EquantHeuristics_2EIS__REMOVABLE__QUANT__FUN,type,
c_2EquantHeuristics_2EIS__REMOVABLE__QUANT__FUN: del > del > \$i ).

thf(mem_c_2EquantHeuristics_2EIS__REMOVABLE__QUANT__FUN,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EquantHeuristics_2EIS__REMOVABLE__QUANT__FUN @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ).

thf(tp_c_2EquantHeuristics_2ESIMPLE__GUESS__EXISTS,type,
c_2EquantHeuristics_2ESIMPLE__GUESS__EXISTS: del > \$i ).

thf(mem_c_2EquantHeuristics_2ESIMPLE__GUESS__EXISTS,axiom,
! [A_27a: del] : ( mem @ ( c_2EquantHeuristics_2ESIMPLE__GUESS__EXISTS @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ bool @ bool ) ) ) ) ).

thf(tp_c_2EquantHeuristics_2ESIMPLE__GUESS__FORALL,type,
c_2EquantHeuristics_2ESIMPLE__GUESS__FORALL: del > \$i ).

thf(mem_c_2EquantHeuristics_2ESIMPLE__GUESS__FORALL,axiom,
! [A_27a: del] : ( mem @ ( c_2EquantHeuristics_2ESIMPLE__GUESS__FORALL @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ bool @ bool ) ) ) ) ).

thf(ax_thm_2EquantHeuristics_2EGUESS__EXISTS__def,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ( ? [V2v: \$i] :
( ( mem @ V2v @ A_27b )
& ( p @ ( ap @ V1P @ V2v ) ) )
<=> ? [V3fv: \$i] :
( ( mem @ V3fv @ A_27a )
& ( p @ ( ap @ V1P @ ( ap @ V0i @ V3fv ) ) ) ) ) ) ) ) ).

thf(ax_thm_2EquantHeuristics_2EGUESS__FORALL__def,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ( ! [V2v: \$i] :
( ( mem @ V2v @ A_27b )
=> ( p @ ( ap @ V1P @ V2v ) ) )
<=> ! [V3fv: \$i] :
( ( mem @ V3fv @ A_27a )
=> ( p @ ( ap @ V1P @ ( ap @ V0i @ V3fv ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__EXISTS__FORALL__REWRITES,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V2v: \$i] :
( ( mem @ V2v @ A_27b )
=> ( ( p @ ( ap @ V1P @ V2v ) )
=> ? [V3fv: \$i] :
( ( mem @ V3fv @ A_27a )
& ( p @ ( ap @ V1P @ ( ap @ V0i @ V3fv ) ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V4v: \$i] :
( ( mem @ V4v @ A_27b )
=> ( ~ ( p @ ( ap @ V1P @ V4v ) )
=> ? [V5fv: \$i] :
( ( mem @ V5fv @ A_27a )
& ~ ( p @ ( ap @ V1P @ ( ap @ V0i @ V5fv ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2EquantHeuristics_2EGUESS__EXISTS__POINT__def,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V2fv: \$i] :
( ( mem @ V2fv @ A_27a )
=> ( p @ ( ap @ V1P @ ( ap @ V0i @ V2fv ) ) ) ) ) ) ) ).

thf(ax_thm_2EquantHeuristics_2EGUESS__FORALL__POINT__def,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V2fv: \$i] :
( ( mem @ V2fv @ A_27a )
=> ~ ( p @ ( ap @ V1P @ ( ap @ V0i @ V2fv ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__POINT__THM,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( ? [V2v: \$i] :
( ( mem @ V2v @ A_27b )
& ( p @ ( ap @ V1P @ V2v ) ) )
<=> \$true ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( ! [V3v: \$i] :
( ( mem @ V3v @ A_27b )
=> ( p @ ( ap @ V1P @ V3v ) ) )
<=> \$false ) ) ) ) ) ).

thf(ax_thm_2EquantHeuristics_2EGUESS__EXISTS__GAP__def,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V2v: \$i] :
( ( mem @ V2v @ A_27b )
=> ( ( p @ ( ap @ V1P @ V2v ) )
=> ? [V3fv: \$i] :
( ( mem @ V3fv @ A_27a )
& ( V2v
= ( ap @ V0i @ V3fv ) ) ) ) ) ) ) ) ).

thf(ax_thm_2EquantHeuristics_2EGUESS__FORALL__GAP__def,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V2v: \$i] :
( ( mem @ V2v @ A_27b )
=> ( ~ ( p @ ( ap @ V1P @ V2v ) )
=> ? [V3fv: \$i] :
( ( mem @ V3fv @ A_27a )
& ( V2v
= ( ap @ V0i @ V3fv ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__REWRITES,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V2v: \$i] :
( ( mem @ V2v @ A_27b )
=> ( ( p @ ( ap @ V1P @ V2v ) )
=> ? [V3fv: \$i] :
( ( mem @ V3fv @ A_27a )
& ( p @ ( ap @ V1P @ ( ap @ V0i @ V3fv ) ) ) ) ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
<=> ! [V4v: \$i] :
( ( mem @ V4v @ A_27b )
=> ( ~ ( p @ ( ap @ V1P @ V4v ) )
=> ? [V5fv: \$i] :
( ( mem @ V5fv @ A_27a )
& ~ ( p @ ( ap @ V1P @ ( ap @ V0i @ V5fv ) ) ) ) ) ) )
& ! [V6i: \$i] :
( ( mem @ V6i @ ( arr @ A_27a @ A_27b ) )
=> ! [V7P: \$i] :
( ( mem @ V7P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V6i ) @ V7P ) )
<=> ! [V8fv: \$i] :
( ( mem @ V8fv @ A_27a )
=> ( p @ ( ap @ V7P @ ( ap @ V6i @ V8fv ) ) ) ) ) ) )
& ! [V9i: \$i] :
( ( mem @ V9i @ ( arr @ A_27a @ A_27b ) )
=> ! [V10P: \$i] :
( ( mem @ V10P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V9i ) @ V10P ) )
<=> ! [V11fv: \$i] :
( ( mem @ V11fv @ A_27a )
=> ~ ( p @ ( ap @ V10P @ ( ap @ V9i @ V11fv ) ) ) ) ) ) )
& ! [V12i: \$i] :
( ( mem @ V12i @ ( arr @ A_27a @ A_27b ) )
=> ! [V13P: \$i] :
( ( mem @ V13P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V12i ) @ V13P ) )
<=> ! [V14v: \$i] :
( ( mem @ V14v @ A_27b )
=> ( ( p @ ( ap @ V13P @ V14v ) )
=> ? [V15fv: \$i] :
( ( mem @ V15fv @ A_27a )
& ( V14v
= ( ap @ V12i @ V15fv ) ) ) ) ) ) ) )
& ! [V16i: \$i] :
( ( mem @ V16i @ ( arr @ A_27a @ A_27b ) )
=> ! [V17P: \$i] :
( ( mem @ V17P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V16i ) @ V17P ) )
<=> ! [V18v: \$i] :
( ( mem @ V18v @ A_27b )
=> ( ~ ( p @ ( ap @ V17P @ V18v ) )
=> ? [V19fv: \$i] :
( ( mem @ V19fv @ A_27a )
& ( V18v
= ( ap @ V16i @ V19fv ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__EXISTS__POINT__THM,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27b ) @ V1P ) )
<=> \$true ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__FORALL__POINT__THM,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27b ) @ V1P ) )
<=> \$false ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__EXISTS__THM,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27b ) @ V1P ) )
<=> ? [V2fv: \$i] :
( ( mem @ V2fv @ A_27a )
& ( p @ ( ap @ V1P @ ( ap @ V0i @ V2fv ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__FORALL__THM,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27b ) @ V1P ) )
<=> ! [V2fv: \$i] :
( ( mem @ V2fv @ A_27a )
=> ( p @ ( ap @ V1P @ ( ap @ V0i @ V2fv ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESSES__UEXISTS__THM1,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ A_27a )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27b @ A_27a )
@ ( lam @ A_27b
@ ^ [V2x: \$i] : V0i ) )
@ V1P ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V1P ) )
<=> ( ( p @ ( ap @ V1P @ V0i ) )
& ! [V3v: \$i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ V1P @ V3v ) )
=> ( V3v = V0i ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESSES__UEXISTS__THM2,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ A_27a )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27a )
@ ( lam @ A_27b
@ ^ [V2x: \$i] : V0i ) )
@ V1P ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V1P ) )
<=> ( p @ ( ap @ V1P @ V0i ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESSES__UEXISTS__THM3,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ A_27a )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27a )
@ ( lam @ A_27b
@ ^ [V2x: \$i] : V0i ) )
@ V1P ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V1P ) )
<=> ! [V3v: \$i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ V1P @ V3v ) )
=> ( V3v = V0i ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESSES__UEXISTS__THM4,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0i: \$i] :
( ( mem @ V0i @ A_27a )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27a )
@ ( lam @ A_27b
@ ^ [V2x: \$i] : V0i ) )
@ V1P ) )
=> ( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27c @ A_27a )
@ ( lam @ A_27c
@ ^ [V3x: \$i] : V0i ) )
@ V1P ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V1P ) )
<=> \$true ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESSES__NEG__DUALITY,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V1P ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V1P ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V1P ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V1P ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V1P ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ bool @ bool ) @ c_2Ebool_2E_7E ) @ V1P ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESSES__NEG__REWRITE,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V2x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V2x ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V3x: \$i] : ( ap @ V1P @ V3x ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V4x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V4x ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V5x: \$i] : ( ap @ V1P @ V5x ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V6x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V6x ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V7x: \$i] : ( ap @ V1P @ V7x ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V8x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V8x ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V9x: \$i] : ( ap @ V1P @ V9x ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V10x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V10x ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V11x: \$i] : ( ap @ V1P @ V11x ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V12x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V12x ) ) ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V13x: \$i] : ( ap @ V1P @ V13x ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESSES__WEAKEN__THM,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i ) @ V1P ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__EQUATION__EXISTS__POINT,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ A_27a )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ A_27b ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( ap @ V1P @ V0i )
= ( ap @ V2Q @ V0i ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ ty_2Eone_2Eone @ A_27a )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V3xxx: \$i] : V0i ) )
@ ( lam @ A_27a
@ ^ [V4x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V1P @ V4x ) ) @ ( ap @ V2Q @ V4x ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__EQUATION__FORALL__POINT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ A_27c ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27b @ A_27c ) )
=> ( ! [V3fv: \$i] :
( ( mem @ V3fv @ A_27a )
=> ( ( ap @ V1P @ ( ap @ V0i @ V3fv ) )
!= ( ap @ V2Q @ ( ap @ V0i @ V3fv ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V4x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27c ) @ ( ap @ V1P @ V4x ) ) @ ( ap @ V2Q @ V4x ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__EQUATION__EXISTS__GAP,axiom,
! [A_27a: del,V0i: \$i] :
( ( mem @ V0i @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ ty_2Eone_2Eone @ A_27a )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V1xxx: \$i] : V0i ) )
@ ( lam @ A_27a
@ ^ [V2x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V0i ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__TRIVIAL__EXISTS__POINT,axiom,
! [A_27a: del,V0i: \$i] :
( ( mem @ V0i @ A_27a )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V1P @ V0i ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ ty_2Eone_2Eone @ A_27a )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V2xxx: \$i] : V0i ) )
@ V1P ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__TRIVIAL__FORALL__POINT,axiom,
! [A_27a: del,V0i: \$i] :
( ( mem @ V0i @ A_27a )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ V1P @ V0i ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ ty_2Eone_2Eone @ A_27a )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V2xxx: \$i] : V0i ) )
@ V1P ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__BOOL,axiom,
( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ ty_2Eone_2Eone @ bool )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V0ARB: \$i] : c_2Ebool_2ET ) )
@ ( lam @ bool
@ ^ [V1x: \$i] : V1x ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ ty_2Eone_2Eone @ bool )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V2ARB: \$i] : c_2Ebool_2EF ) )
@ ( lam @ bool
@ ^ [V3x: \$i] : V3x ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ ty_2Eone_2Eone @ bool )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V4ARB: \$i] : c_2Ebool_2ET ) )
@ ( lam @ bool
@ ^ [V5x: \$i] : V5x ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ ty_2Eone_2Eone @ bool )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V6ARB: \$i] : c_2Ebool_2EF ) )
@ ( lam @ bool
@ ^ [V7x: \$i] : V7x ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__TWO__CASES,axiom,
! [A_27a: del,A_27b: del,V0y: \$i] :
( ( mem @ V0y @ A_27a )
=> ! [V1Q: \$i] :
( ( mem @ V1Q @ ( arr @ A_27b @ A_27a ) )
=> ( ! [V2x: \$i] :
( ( mem @ V2x @ A_27a )
=> ( ( V2x = V0y )
| ? [V3fv: \$i] :
( ( mem @ V3fv @ A_27b )
& ( V2x
= ( ap @ V1Q @ V3fv ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27a ) @ V1Q )
@ ( lam @ A_27a
@ ^ [V4x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V4x ) @ V0y ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__ONE__CASE______FORALL__GAP,axiom,
! [A_27a: del,A_27b: del,V0P: \$i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: \$i] :
( ( mem @ V1Q @ ( arr @ A_27b @ A_27a ) )
=> ( ! [V2x: \$i] :
( ( mem @ V2x @ A_27a )
=> ? [V3fv: \$i] :
( ( mem @ V3fv @ A_27b )
& ( V2x
= ( ap @ V1Q @ V3fv ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27a ) @ V1Q ) @ V0P ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__ONE__CASE______EXISTS__GAP,axiom,
! [A_27a: del,A_27b: del,V0P: \$i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: \$i] :
( ( mem @ V1Q @ ( arr @ A_27b @ A_27a ) )
=> ( ! [V2x: \$i] :
( ( mem @ V2x @ A_27a )
=> ? [V3fv: \$i] :
( ( mem @ V3fv @ A_27b )
& ( V2x
= ( ap @ V1Q @ V3fv ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27a ) @ V1Q ) @ V0P ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__NEG,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V2x: \$i] : ( ap @ V1P @ V2x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V3x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V3x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V4x: \$i] : ( ap @ V1P @ V4x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V5x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V5x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V6x: \$i] : ( ap @ V1P @ V6x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V7x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V7x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V8x: \$i] : ( ap @ V1P @ V8x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V9x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V9x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V10x: \$i] : ( ap @ V1P @ V10x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V11x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V11x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V12x: \$i] : ( ap @ V1P @ V12x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V13x: \$i] : ( ap @ c_2Ebool_2E_7E @ ( ap @ V1P @ V13x ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__CONSTANT__EXISTS,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1p: \$i] :
( ( mem @ V1p @ bool )
=> ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V2x: \$i] : V1p ) ) )
<=> \$true ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__CONSTANT__FORALL,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1p: \$i] :
( ( mem @ V1p @ bool )
=> ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V2x: \$i] : V1p ) ) )
<=> \$true ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__DISJ,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ! [V3iK: \$i] :
( ( mem @ V3iK @ A_27b )
=> ! [V4q: \$i] :
( ( mem @ V4q @ bool )
=> ! [V5p: \$i] :
( ( mem @ V5p @ bool )
=> ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V6x: \$i] : ( ap @ V1P @ V6x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V7x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V7x ) ) @ ( ap @ V2Q @ V7x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V8x: \$i] : ( ap @ V2Q @ V8x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V9x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V9x ) ) @ ( ap @ V2Q @ V9x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V10x: \$i] : ( ap @ V1P @ V10x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V11x: \$i] : ( ap @ V2Q @ V11x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V12x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V12x ) ) @ ( ap @ V2Q @ V12x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V13x: \$i] : ( ap @ V1P @ V13x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V14x: \$i] : ( ap @ V2Q @ V14x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V15x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V15x ) ) @ ( ap @ V2Q @ V15x ) ) ) ) ) )
& ( ( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V16xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V17x: \$i] : ( ap @ V1P @ V17x ) ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V18xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V19x: \$i] : ( ap @ V2Q @ V19x ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V20xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V21x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V21x ) ) @ ( ap @ V2Q @ V21x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V22x: \$i] : ( ap @ V1P @ V22x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V23x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V23x ) ) @ V4q ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V24x: \$i] : ( ap @ V2Q @ V24x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V25x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ V5p ) @ ( ap @ V2Q @ V25x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V26x: \$i] : ( ap @ V1P @ V26x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V27x: \$i] : ( ap @ V2Q @ V27x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V28x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V28x ) ) @ ( ap @ V2Q @ V28x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V29x: \$i] : ( ap @ V1P @ V29x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V30x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V30x ) ) @ ( ap @ V2Q @ V30x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V31x: \$i] : ( ap @ V2Q @ V31x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V32x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ V1P @ V32x ) ) @ ( ap @ V2Q @ V32x ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__CONJ,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ! [V3iK: \$i] :
( ( mem @ V3iK @ A_27b )
=> ! [V4q: \$i] :
( ( mem @ V4q @ bool )
=> ! [V5p: \$i] :
( ( mem @ V5p @ bool )
=> ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V6x: \$i] : ( ap @ V1P @ V6x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V7x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V7x ) ) @ ( ap @ V2Q @ V7x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V8x: \$i] : ( ap @ V2Q @ V8x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V9x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V9x ) ) @ ( ap @ V2Q @ V9x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V10x: \$i] : ( ap @ V1P @ V10x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V11x: \$i] : ( ap @ V2Q @ V11x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V12x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V12x ) ) @ ( ap @ V2Q @ V12x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V13x: \$i] : ( ap @ V1P @ V13x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V14x: \$i] : ( ap @ V2Q @ V14x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V15x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V15x ) ) @ ( ap @ V2Q @ V15x ) ) ) ) ) )
& ( ( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V16xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V17x: \$i] : ( ap @ V1P @ V17x ) ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V18xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V19x: \$i] : ( ap @ V2Q @ V19x ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V20xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V21x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V21x ) ) @ ( ap @ V2Q @ V21x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V22x: \$i] : ( ap @ V1P @ V22x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V23x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V23x ) ) @ V4q ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V24x: \$i] : ( ap @ V2Q @ V24x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V25x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ V5p ) @ ( ap @ V2Q @ V25x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V26x: \$i] : ( ap @ V1P @ V26x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V27x: \$i] : ( ap @ V2Q @ V27x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V28x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V28x ) ) @ ( ap @ V2Q @ V28x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V29x: \$i] : ( ap @ V1P @ V29x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V30x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V30x ) ) @ ( ap @ V2Q @ V30x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V31x: \$i] : ( ap @ V2Q @ V31x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V32x: \$i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V32x ) ) @ ( ap @ V2Q @ V32x ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__IMP,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ! [V3iK: \$i] :
( ( mem @ V3iK @ A_27b )
=> ! [V4q: \$i] :
( ( mem @ V4q @ bool )
=> ! [V5p: \$i] :
( ( mem @ V5p @ bool )
=> ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V6x: \$i] : ( ap @ V1P @ V6x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V7x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V7x ) ) @ ( ap @ V2Q @ V7x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V8x: \$i] : ( ap @ V2Q @ V8x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V9x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V9x ) ) @ ( ap @ V2Q @ V9x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V10x: \$i] : ( ap @ V1P @ V10x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V11x: \$i] : ( ap @ V2Q @ V11x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V12x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V12x ) ) @ ( ap @ V2Q @ V12x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V13x: \$i] : ( ap @ V1P @ V13x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V14x: \$i] : ( ap @ V2Q @ V14x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V15x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V15x ) ) @ ( ap @ V2Q @ V15x ) ) ) ) ) )
& ( ( ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V16xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V17x: \$i] : ( ap @ V1P @ V17x ) ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V18xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V19x: \$i] : ( ap @ V2Q @ V19x ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ty_2Eone_2Eone @ A_27b )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V20xxx: \$i] : V3iK ) )
@ ( lam @ A_27b
@ ^ [V21x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V21x ) ) @ ( ap @ V2Q @ V21x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V22x: \$i] : ( ap @ V1P @ V22x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V23x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V23x ) ) @ V4q ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V24x: \$i] : ( ap @ V2Q @ V24x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V25x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ V5p ) @ ( ap @ V2Q @ V25x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V26x: \$i] : ( ap @ V1P @ V26x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V27x: \$i] : ( ap @ V2Q @ V27x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V28x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V28x ) ) @ ( ap @ V2Q @ V28x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V29x: \$i] : ( ap @ V1P @ V29x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V30x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V30x ) ) @ ( ap @ V2Q @ V30x ) ) ) ) ) )
& ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V31x: \$i] : ( ap @ V2Q @ V31x ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V32x: \$i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1P @ V32x ) ) @ ( ap @ V2Q @ V32x ) ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__EQUIV,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ! [V3P1: \$i] :
( ( mem @ V3P1 @ ( arr @ A_27b @ bool ) )
=> ! [V4P2: \$i] :
( ( mem @ V4P2 @ ( arr @ A_27b @ bool ) )
=> ( ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V5x: \$i] : ( ap @ V1P @ V5x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V6x: \$i] : ( ap @ V2Q @ V6x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V7x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V1P @ V7x ) ) @ ( ap @ V2Q @ V7x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V8x: \$i] : ( ap @ V1P @ V8x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V9x: \$i] : ( ap @ V2Q @ V9x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V10x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V1P @ V10x ) ) @ ( ap @ V2Q @ V10x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V11x: \$i] : ( ap @ V1P @ V11x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V12x: \$i] : ( ap @ V2Q @ V12x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V13x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V1P @ V13x ) ) @ ( ap @ V2Q @ V13x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V14x: \$i] : ( ap @ V1P @ V14x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V15x: \$i] : ( ap @ V2Q @ V15x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V16x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V1P @ V16x ) ) @ ( ap @ V2Q @ V16x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V17x: \$i] : ( ap @ V3P1 @ V17x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V18x: \$i] : ( ap @ V4P2 @ V18x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V19x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V3P1 @ V19x ) ) @ ( ap @ V4P2 @ V19x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V20x: \$i] : ( ap @ V3P1 @ V20x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V21x: \$i] : ( ap @ V4P2 @ V21x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V22x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V3P1 @ V22x ) ) @ ( ap @ V4P2 @ V22x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V23x: \$i] : ( ap @ V3P1 @ V23x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V24x: \$i] : ( ap @ V4P2 @ V24x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V25x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V3P1 @ V25x ) ) @ ( ap @ V4P2 @ V25x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V26x: \$i] : ( ap @ V3P1 @ V26x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V27x: \$i] : ( ap @ V4P2 @ V27x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V28x: \$i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ V3P1 @ V28x ) ) @ ( ap @ V4P2 @ V28x ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__COND,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27b ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27b @ bool ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ! [V3b: \$i] :
( ( mem @ V3b @ ( arr @ A_27b @ bool ) )
=> ! [V4bc: \$i] :
( ( mem @ V4bc @ bool )
=> ( ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V5x: \$i] : ( ap @ V1P @ V5x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V6x: \$i] : ( ap @ V2Q @ V6x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V7x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V7x ) ) @ ( ap @ V1P @ V7x ) ) @ ( ap @ V2Q @ V7x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V8x: \$i] : ( ap @ V1P @ V8x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V9x: \$i] : ( ap @ V2Q @ V9x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V10x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V10x ) ) @ ( ap @ V1P @ V10x ) ) @ ( ap @ V2Q @ V10x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V11x: \$i] : ( ap @ V1P @ V11x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V12x: \$i] : ( ap @ V2Q @ V12x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V13x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V4bc ) @ ( ap @ V1P @ V13x ) ) @ ( ap @ V2Q @ V13x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V14x: \$i] : ( ap @ V1P @ V14x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V15x: \$i] : ( ap @ V2Q @ V15x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V16x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V4bc ) @ ( ap @ V1P @ V16x ) ) @ ( ap @ V2Q @ V16x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V17x: \$i] : ( ap @ V1P @ V17x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V18x: \$i] : ( ap @ V2Q @ V18x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V19x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V19x ) ) @ ( ap @ V1P @ V19x ) ) @ ( ap @ V2Q @ V19x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V20x: \$i] : ( ap @ V1P @ V20x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V21x: \$i] : ( ap @ V2Q @ V21x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V22x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V22x ) ) @ ( ap @ V1P @ V22x ) ) @ ( ap @ V2Q @ V22x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V23x: \$i] : ( ap @ V3b @ V23x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V24x: \$i] : ( ap @ V2Q @ V24x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V25x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V25x ) ) @ ( ap @ V1P @ V25x ) ) @ ( ap @ V2Q @ V25x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V26x: \$i] : ( ap @ V3b @ V26x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V27x: \$i] : ( ap @ V2Q @ V27x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V28x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V28x ) ) @ ( ap @ V1P @ V28x ) ) @ ( ap @ V2Q @ V28x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V29x: \$i] : ( ap @ V3b @ V29x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V30x: \$i] : ( ap @ V1P @ V30x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V31x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V31x ) ) @ ( ap @ V1P @ V31x ) ) @ ( ap @ V2Q @ V31x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V32x: \$i] : ( ap @ V3b @ V32x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V33x: \$i] : ( ap @ V1P @ V33x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V34x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V34x ) ) @ ( ap @ V1P @ V34x ) ) @ ( ap @ V2Q @ V34x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V35x: \$i] : ( ap @ V3b @ V35x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V36x: \$i] : ( ap @ V1P @ V36x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V37x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V37x ) ) @ ( ap @ V1P @ V37x ) ) @ ( ap @ V2Q @ V37x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V38x: \$i] : ( ap @ V3b @ V38x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V39x: \$i] : ( ap @ V2Q @ V39x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V40x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V40x ) ) @ ( ap @ V1P @ V40x ) ) @ ( ap @ V2Q @ V40x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V41x: \$i] : ( ap @ V3b @ V41x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V42x: \$i] : ( ap @ V2Q @ V42x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V43x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V43x ) ) @ ( ap @ V1P @ V43x ) ) @ ( ap @ V2Q @ V43x ) ) ) ) ) )
& ( ( ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V44x: \$i] : ( ap @ V3b @ V44x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V45x: \$i] : ( ap @ V1P @ V45x ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b ) @ V0i )
@ ( lam @ A_27b
@ ^ [V46x: \$i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ ( ap @ V3b @ V46x ) ) @ ( ap @ V1P @ V46x ) ) @ ( ap @ V2Q @ V46x ) ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__FORALL______NEW__FV,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0iy: \$i] :
( ( mem @ V0iy @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2y: \$i] :
( ( mem @ V2y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27c ) @ ( ap @ V0iy @ V2y ) )
@ ( lam @ A_27c
@ ^ [V3x: \$i] : ( ap @ ( ap @ V1P @ V3x ) @ V2y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V4fv: \$i] : ( ap @ ( ap @ V0iy @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V4fv ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V4fv ) ) ) )
@ ( lam @ A_27c
@ ^ [V5x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V6y: \$i] : ( ap @ ( ap @ V1P @ V5x ) @ V6y ) ) ) ) ) ) )
& ( ! [V7y: \$i] :
( ( mem @ V7y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27b @ A_27c ) @ ( ap @ V0iy @ V7y ) )
@ ( lam @ A_27c
@ ^ [V8x: \$i] : ( ap @ ( ap @ V1P @ V8x ) @ V7y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V9fv: \$i] : ( ap @ ( ap @ V0iy @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V9fv ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V9fv ) ) ) )
@ ( lam @ A_27c
@ ^ [V10x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V11y: \$i] : ( ap @ ( ap @ V1P @ V10x ) @ V11y ) ) ) ) ) ) )
& ( ! [V12y: \$i] :
( ( mem @ V12y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27c ) @ ( ap @ V0iy @ V12y ) )
@ ( lam @ A_27c
@ ^ [V13x: \$i] : ( ap @ ( ap @ V1P @ V13x ) @ V12y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V14fv: \$i] : ( ap @ ( ap @ V0iy @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V14fv ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V14fv ) ) ) )
@ ( lam @ A_27c
@ ^ [V15x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V16y: \$i] : ( ap @ ( ap @ V1P @ V15x ) @ V16y ) ) ) ) ) ) )
& ( ! [V17y: \$i] :
( ( mem @ V17y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27c ) @ ( ap @ V0iy @ V17y ) )
@ ( lam @ A_27c
@ ^ [V18x: \$i] : ( ap @ ( ap @ V1P @ V18x ) @ V17y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V19fv: \$i] : ( ap @ ( ap @ V0iy @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V19fv ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V19fv ) ) ) )
@ ( lam @ A_27c
@ ^ [V20x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V21y: \$i] : ( ap @ ( ap @ V1P @ V20x ) @ V21y ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__FORALL______NEW__FV__1,axiom,
! [A_27a: del,A_27c: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27c ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2y: \$i] :
( ( mem @ V2y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V3xxx: \$i] : ( ap @ V0i @ V2y ) ) )
@ ( lam @ A_27c
@ ^ [V4x: \$i] : ( ap @ ( ap @ V1P @ V4x ) @ V2y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V5x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V6y: \$i] : ( ap @ ( ap @ V1P @ V5x ) @ V6y ) ) ) ) ) ) )
& ( ! [V7y: \$i] :
( ( mem @ V7y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V8xxx: \$i] : ( ap @ V0i @ V7y ) ) )
@ ( lam @ A_27c
@ ^ [V9x: \$i] : ( ap @ ( ap @ V1P @ V9x ) @ V7y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V10x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V11y: \$i] : ( ap @ ( ap @ V1P @ V10x ) @ V11y ) ) ) ) ) ) )
& ( ! [V12y: \$i] :
( ( mem @ V12y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V13xxx: \$i] : ( ap @ V0i @ V12y ) ) )
@ ( lam @ A_27c
@ ^ [V14x: \$i] : ( ap @ ( ap @ V1P @ V14x ) @ V12y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V15x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V16y: \$i] : ( ap @ ( ap @ V1P @ V15x ) @ V16y ) ) ) ) ) ) )
& ( ! [V17y: \$i] :
( ( mem @ V17y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V18xxx: \$i] : ( ap @ V0i @ V17y ) ) )
@ ( lam @ A_27c
@ ^ [V19x: \$i] : ( ap @ ( ap @ V1P @ V19x ) @ V17y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V20x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V21y: \$i] : ( ap @ ( ap @ V1P @ V20x ) @ V21y ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__FORALL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27b @ A_27c ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ! [V2iK: \$i] :
( ( mem @ V2iK @ A_27c )
=> ( ( ! [V3y: \$i] :
( ( mem @ V3y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V4x: \$i] : ( ap @ ( ap @ V1P @ V4x ) @ V3y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V5x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V6y: \$i] : ( ap @ ( ap @ V1P @ V5x ) @ V6y ) ) ) ) ) ) )
& ( ! [V7y: \$i] :
( ( mem @ V7y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V8x: \$i] : ( ap @ ( ap @ V1P @ V8x ) @ V7y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V9x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V10y: \$i] : ( ap @ ( ap @ V1P @ V9x ) @ V10y ) ) ) ) ) ) )
& ( ! [V11y: \$i] :
( ( mem @ V11y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V12x: \$i] : ( ap @ ( ap @ V1P @ V12x ) @ V11y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V13x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V14y: \$i] : ( ap @ ( ap @ V1P @ V13x ) @ V14y ) ) ) ) ) ) )
& ( ! [V15y: \$i] :
( ( mem @ V15y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V16x: \$i] : ( ap @ ( ap @ V1P @ V16x ) @ V15y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V17x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V18y: \$i] : ( ap @ ( ap @ V1P @ V17x ) @ V18y ) ) ) ) ) ) )
& ( ! [V19y: \$i] :
( ( mem @ V19y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V20xxx: \$i] : V2iK ) )
@ ( lam @ A_27c
@ ^ [V21x: \$i] : ( ap @ ( ap @ V1P @ V21x ) @ V19y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V22xxx: \$i] : V2iK ) )
@ ( lam @ A_27c
@ ^ [V23x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V24y: \$i] : ( ap @ ( ap @ V1P @ V23x ) @ V24y ) ) ) ) ) ) )
& ( ! [V25y: \$i] :
( ( mem @ V25y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V26x: \$i] : ( ap @ ( ap @ V1P @ V26x ) @ V25y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V27x: \$i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V28y: \$i] : ( ap @ ( ap @ V1P @ V27x ) @ V28y ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__EXISTS______NEW__FV,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0iy: \$i] :
( ( mem @ V0iy @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2y: \$i] :
( ( mem @ V2y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27c ) @ ( ap @ V0iy @ V2y ) )
@ ( lam @ A_27c
@ ^ [V3x: \$i] : ( ap @ ( ap @ V1P @ V3x ) @ V2y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V4fv: \$i] : ( ap @ ( ap @ V0iy @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V4fv ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V4fv ) ) ) )
@ ( lam @ A_27c
@ ^ [V5x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6y: \$i] : ( ap @ ( ap @ V1P @ V5x ) @ V6y ) ) ) ) ) ) )
& ( ! [V7y: \$i] :
( ( mem @ V7y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27b @ A_27c ) @ ( ap @ V0iy @ V7y ) )
@ ( lam @ A_27c
@ ^ [V8x: \$i] : ( ap @ ( ap @ V1P @ V8x ) @ V7y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V9fv: \$i] : ( ap @ ( ap @ V0iy @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V9fv ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V9fv ) ) ) )
@ ( lam @ A_27c
@ ^ [V10x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V11y: \$i] : ( ap @ ( ap @ V1P @ V10x ) @ V11y ) ) ) ) ) ) )
& ( ! [V12y: \$i] :
( ( mem @ V12y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27c ) @ ( ap @ V0iy @ V12y ) )
@ ( lam @ A_27c
@ ^ [V13x: \$i] : ( ap @ ( ap @ V1P @ V13x ) @ V12y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V14fv: \$i] : ( ap @ ( ap @ V0iy @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V14fv ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V14fv ) ) ) )
@ ( lam @ A_27c
@ ^ [V15x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V16y: \$i] : ( ap @ ( ap @ V1P @ V15x ) @ V16y ) ) ) ) ) ) )
& ( ! [V17y: \$i] :
( ( mem @ V17y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27c ) @ ( ap @ V0iy @ V17y ) )
@ ( lam @ A_27c
@ ^ [V18x: \$i] : ( ap @ ( ap @ V1P @ V18x ) @ V17y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V19fv: \$i] : ( ap @ ( ap @ V0iy @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V19fv ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V19fv ) ) ) )
@ ( lam @ A_27c
@ ^ [V20x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V21y: \$i] : ( ap @ ( ap @ V1P @ V20x ) @ V21y ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__EXISTS______NEW__FV__1,axiom,
! [A_27a: del,A_27c: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27a @ A_27c ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2y: \$i] :
( ( mem @ V2y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V3xxx: \$i] : ( ap @ V0i @ V2y ) ) )
@ ( lam @ A_27c
@ ^ [V4x: \$i] : ( ap @ ( ap @ V1P @ V4x ) @ V2y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V5x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6y: \$i] : ( ap @ ( ap @ V1P @ V5x ) @ V6y ) ) ) ) ) ) )
& ( ! [V7y: \$i] :
( ( mem @ V7y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V8xxx: \$i] : ( ap @ V0i @ V7y ) ) )
@ ( lam @ A_27c
@ ^ [V9x: \$i] : ( ap @ ( ap @ V1P @ V9x ) @ V7y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V10x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V11y: \$i] : ( ap @ ( ap @ V1P @ V10x ) @ V11y ) ) ) ) ) ) )
& ( ! [V12y: \$i] :
( ( mem @ V12y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V13xxx: \$i] : ( ap @ V0i @ V12y ) ) )
@ ( lam @ A_27c
@ ^ [V14x: \$i] : ( ap @ ( ap @ V1P @ V14x ) @ V12y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V15x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V16y: \$i] : ( ap @ ( ap @ V1P @ V15x ) @ V16y ) ) ) ) ) ) )
& ( ! [V17y: \$i] :
( ( mem @ V17y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V18xxx: \$i] : ( ap @ V0i @ V17y ) ) )
@ ( lam @ A_27c
@ ^ [V19x: \$i] : ( ap @ ( ap @ V1P @ V19x ) @ V17y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V20x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V21y: \$i] : ( ap @ ( ap @ V1P @ V20x ) @ V21y ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__EXISTS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27b @ A_27c ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ! [V2iK: \$i] :
( ( mem @ V2iK @ A_27c )
=> ( ( ! [V3y: \$i] :
( ( mem @ V3y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V4x: \$i] : ( ap @ ( ap @ V1P @ V4x ) @ V3y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V5x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6y: \$i] : ( ap @ ( ap @ V1P @ V5x ) @ V6y ) ) ) ) ) ) )
& ( ! [V7y: \$i] :
( ( mem @ V7y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V8x: \$i] : ( ap @ ( ap @ V1P @ V8x ) @ V7y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V9x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V10y: \$i] : ( ap @ ( ap @ V1P @ V9x ) @ V10y ) ) ) ) ) ) )
& ( ! [V11y: \$i] :
( ( mem @ V11y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V12x: \$i] : ( ap @ ( ap @ V1P @ V12x ) @ V11y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V13x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V14y: \$i] : ( ap @ ( ap @ V1P @ V13x ) @ V14y ) ) ) ) ) ) )
& ( ! [V15y: \$i] :
( ( mem @ V15y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V16x: \$i] : ( ap @ ( ap @ V1P @ V16x ) @ V15y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V17x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V18y: \$i] : ( ap @ ( ap @ V1P @ V17x ) @ V18y ) ) ) ) ) ) )
& ( ! [V19y: \$i] :
( ( mem @ V19y @ A_27a )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V20xxx: \$i] : V2iK ) )
@ ( lam @ A_27c
@ ^ [V21x: \$i] : ( ap @ ( ap @ V1P @ V21x ) @ V19y ) ) ) ) )
=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ty_2Eone_2Eone @ A_27c )
@ ( lam @ ty_2Eone_2Eone
@ ^ [V22xxx: \$i] : V2iK ) )
@ ( lam @ A_27c
@ ^ [V23x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V24y: \$i] : ( ap @ ( ap @ V1P @ V23x ) @ V24y ) ) ) ) ) ) )
& ( ! [V25y: \$i] :
( ( mem @ V25y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V26x: \$i] : ( ap @ ( ap @ V1P @ V26x ) @ V25y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V27x: \$i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V28y: \$i] : ( ap @ ( ap @ V1P @ V27x ) @ V28y ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__EXISTS__UNIQUE,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27b @ A_27c ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2y: \$i] :
( ( mem @ V2y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V3x: \$i] : ( ap @ ( ap @ V1P @ V3x ) @ V2y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V4x: \$i] :
( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5y: \$i] : ( ap @ ( ap @ V1P @ V4x ) @ V5y ) ) ) ) ) ) )
& ( ! [V6y: \$i] :
( ( mem @ V6y @ A_27a )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V7x: \$i] : ( ap @ ( ap @ V1P @ V7x ) @ V6y ) ) ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27b @ A_27c ) @ V0i )
@ ( lam @ A_27c
@ ^ [V8x: \$i] :
( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V9y: \$i] : ( ap @ ( ap @ V1P @ V8x ) @ V9y ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__ELIM__UNIT,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Eone_2Eone ) @ A_27b ) )
=> ! [V1vt: \$i] :
( ( mem @ V1vt @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Eone_2Eone ) @ A_27b ) @ V0i ) @ V1vt ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V2x: \$i] : ( ap @ V0i @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Eone_2Eone ) @ V2x ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
@ V1vt ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Eone_2Eone ) @ A_27b ) @ V0i ) @ V1vt ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V3x: \$i] : ( ap @ V0i @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Eone_2Eone ) @ V3x ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
@ V1vt ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Eone_2Eone ) @ A_27b ) @ V0i ) @ V1vt ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V4x: \$i] : ( ap @ V0i @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Eone_2Eone ) @ V4x ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
@ V1vt ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Eone_2Eone ) @ A_27b ) @ V0i ) @ V1vt ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V5x: \$i] : ( ap @ V0i @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Eone_2Eone ) @ V5x ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
@ V1vt ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Eone_2Eone ) @ A_27b ) @ V0i ) @ V1vt ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__GAP @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V6x: \$i] : ( ap @ V0i @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Eone_2Eone ) @ V6x ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
@ V1vt ) ) )
& ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ ( ty_2Epair_2Eprod @ A_27a @ ty_2Eone_2Eone ) @ A_27b ) @ V0i ) @ V1vt ) )
<=> ( p
@ ( ap
@ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V7x: \$i] : ( ap @ V0i @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Eone_2Eone ) @ V7x ) @ ( inj__ty_2Eone_2Eone @ fo__c_2Eone_2Eone ) ) ) ) )
@ V1vt ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__STRENGTHEN__EXISTS__POINT,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27b @ A_27a ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: \$i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V1P @ V3x ) )
=> ( p @ ( ap @ V2Q @ V3x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27a ) @ V0i ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__EXISTS__POINT @ A_27b @ A_27a ) @ V0i ) @ V2Q ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__STRENGTHEN__FORALL__GAP,axiom,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27b @ A_27a ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: \$i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V1P @ V3x ) )
=> ( p @ ( ap @ V2Q @ V3x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27a ) @ V0i ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__GAP @ A_27b @ A_27a ) @ V0i ) @ V2Q ) ) ) ) ) ) ) ).

thf(conj_thm_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT,conjecture,
! [A_27a: del,A_27b: del,V0i: \$i] :
( ( mem @ V0i @ ( arr @ A_27b @ A_27a ) )
=> ! [V1P: \$i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: \$i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: \$i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V2Q @ V3x ) )
=> ( p @ ( ap @ V1P @ V3x ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27a ) @ V0i ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2EquantHeuristics_2EGUESS__FORALL__POINT @ A_27b @ A_27a ) @ V0i ) @ V2Q ) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------
```