ITP001 Axioms: ITP073^5.ax
%------------------------------------------------------------------------------
% File : ITP073^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : quotient_sum^2.ax [Gau20]
% : HL4073^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 17 ( 1 unt; 1 typ; 0 def)
% Number of atoms : 732 ( 4 equ; 0 cnn)
% Maximal formula atoms : 107 ( 43 avg)
% Number of connectives : 1528 ( 0 ~; 0 |; 7 &;1354 @)
% ( 8 <=>; 159 =>; 0 <=; 0 <~>)
% Maximal formula depth : 60 ( 31 avg;1354 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 2 ( 2 >; 0 *; 0 +; 0 <<)
% Number of symbols : 20 ( 18 usr; 18 con; 0-2 aty)
% Number of variables : 187 ( 0 ^ 187 !; 0 ?; 187 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Equotient__sum_2E_2B_2B_2B,type,
c_2Equotient__sum_2E_2B_2B_2B: del > del > $i ).
thf(mem_c_2Equotient__sum_2E_2B_2B_2B,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2ESUM__REL__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ bool ) ) ) ) )
=> ( ( ! [V1R1: $i] :
( ( mem @ V1R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2R2: $i] :
( ( mem @ V2R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V3a1: $i] :
( ( mem @ V3a1 @ A_27a )
=> ! [V4a2: $i] :
( ( mem @ V4a2 @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1R1 ) @ V2R2 ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V3a1 ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V4a2 ) ) ) ) ) ) )
& ! [V5R1: $i] :
( ( mem @ V5R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V6R2: $i] :
( ( mem @ V6R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V7b1: $i] :
( ( mem @ V7b1 @ A_27b )
=> ! [V8b2: $i] :
( ( mem @ V8b2 @ A_27b )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V5R1 ) @ V6R2 ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V7b1 ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V8b2 ) ) ) ) ) ) )
& ! [V9R1: $i] :
( ( mem @ V9R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V10R2: $i] :
( ( mem @ V10R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V11a1: $i] :
( ( mem @ V11a1 @ A_27a )
=> ! [V12b2: $i] :
( ( mem @ V12b2 @ A_27b )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V9R1 ) @ V10R2 ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V11a1 ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V12b2 ) ) ) ) ) ) )
& ! [V13R1: $i] :
( ( mem @ V13R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V14R2: $i] :
( ( mem @ V14R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V15b1: $i] :
( ( mem @ V15b1 @ A_27b )
=> ! [V16a2: $i] :
( ( mem @ V16a2 @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V13R1 ) @ V14R2 ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V15b1 ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V16a2 ) ) ) ) ) ) ) )
=> ! [V17v: $i] :
( ( mem @ V17v @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V18v1: $i] :
( ( mem @ V18v1 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V19v2: $i] :
( ( mem @ V19v2 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ! [V20v3: $i] :
( ( mem @ V20v3 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V17v ) @ V18v1 ) @ V19v2 ) @ V20v3 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2ESUM__REL__def,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2a1: $i] :
( ( mem @ V2a1 @ A_27a )
=> ! [V3a2: $i] :
( ( mem @ V3a2 @ A_27a )
=> ! [V4b1: $i] :
( ( mem @ V4b1 @ A_27b )
=> ! [V5b2: $i] :
( ( mem @ V5b2 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2a1 ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V3a2 ) ) )
<=> ( p @ ( ap @ ( ap @ V0R1 @ V2a1 ) @ V3a2 ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V4b1 ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V5b2 ) ) )
<=> ( p @ ( ap @ ( ap @ V1R2 @ V4b1 ) @ V5b2 ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V2a1 ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V5b2 ) ) )
<=> $false )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V4b1 ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V3a2 ) ) )
<=> $false ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2ESUM__REL__EQ,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ ( c_2Emin_2E_3D @ A_27b ) )
= ( c_2Emin_2E_3D @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Equotient__sum_2ESUM__EQUIV,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0R1 ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27b ) @ V1R2 ) )
=> ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2ESUM__QUOTIENT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) @ ( ty_2Esum_2Esum @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d ) @ V1abs1 ) @ V4abs2 ) ) @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27c @ A_27d @ A_27a @ A_27b ) @ V2rep1 ) @ V5rep2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2EINL__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6a: $i] :
( ( mem @ V6a @ A_27c )
=> ( ( ap @ ( c_2Esum_2EINL @ A_27c @ A_27d ) @ V6a )
= ( ap @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d ) @ V1abs1 ) @ V4abs2 ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ ( ap @ V2rep1 @ V6a ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2EINL__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6a1: $i] :
( ( mem @ V6a1 @ A_27a )
=> ! [V7a2: $i] :
( ( mem @ V7a2 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V6a1 ) @ V7a2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V6a1 ) ) @ ( ap @ ( c_2Esum_2EINL @ A_27a @ A_27b ) @ V7a2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2EINR__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6b: $i] :
( ( mem @ V6b @ A_27d )
=> ( ( ap @ ( c_2Esum_2EINR @ A_27c @ A_27d ) @ V6b )
= ( ap @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27b @ A_27c @ A_27d ) @ V1abs1 ) @ V4abs2 ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ ( ap @ V5rep2 @ V6b ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2EINR__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6b1: $i] :
( ( mem @ V6b1 @ A_27b )
=> ! [V7b2: $i] :
( ( mem @ V7b2 @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V3R2 @ V6b1 ) @ V7b2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V6b1 ) ) @ ( ap @ ( c_2Esum_2EINR @ A_27a @ A_27b ) @ V7b2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2EISL__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6a: $i] :
( ( mem @ V6a @ ( ty_2Esum_2Esum @ A_27c @ A_27d ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EISL @ A_27c @ A_27d ) @ V6a ) )
<=> ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27c @ A_27d @ A_27a @ A_27b ) @ V2rep1 ) @ V5rep2 ) @ V6a ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2EISL__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6a1: $i] :
( ( mem @ V6a1 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ! [V7a2: $i] :
( ( mem @ V7a2 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6a1 ) @ V7a2 ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ V6a1 ) )
<=> ( p @ ( ap @ ( c_2Esum_2EISL @ A_27a @ A_27b ) @ V7a2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2EISR__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6a: $i] :
( ( mem @ V6a @ ( ty_2Esum_2Esum @ A_27c @ A_27d ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EISR @ A_27c @ A_27d ) @ V6a ) )
<=> ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27c @ A_27d @ A_27a @ A_27b ) @ V2rep1 ) @ V5rep2 ) @ V6a ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2EISR__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6a1: $i] :
( ( mem @ V6a1 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ! [V7a2: $i] :
( ( mem @ V7a2 @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6a1 ) @ V7a2 ) )
=> ( ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ V6a1 ) )
<=> ( p @ ( ap @ ( c_2Esum_2EISR @ A_27a @ A_27b ) @ V7a2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2ESUM__MAP__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27e ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27e @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27e ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27f ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27f @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27f ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27g ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27g @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27g ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9R4: $i] :
( ( mem @ V9R4 @ ( arr @ A_27d @ ( arr @ A_27d @ bool ) ) )
=> ! [V10abs4: $i] :
( ( mem @ V10abs4 @ ( arr @ A_27d @ A_27h ) )
=> ! [V11rep4: $i] :
( ( mem @ V11rep4 @ ( arr @ A_27h @ A_27d ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27d @ A_27h ) @ V9R4 ) @ V10abs4 ) @ V11rep4 ) )
=> ! [V12f: $i] :
( ( mem @ V12f @ ( arr @ A_27e @ A_27f ) )
=> ! [V13g: $i] :
( ( mem @ V13g @ ( arr @ A_27g @ A_27h ) )
=> ( ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27e @ A_27g @ A_27f @ A_27h ) @ V12f ) @ V13g )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ ( ty_2Esum_2Esum @ A_27e @ A_27g ) @ ( ty_2Esum_2Esum @ A_27b @ A_27d ) @ ( ty_2Esum_2Esum @ A_27a @ A_27c ) @ ( ty_2Esum_2Esum @ A_27f @ A_27h ) ) @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27e @ A_27g @ A_27a @ A_27c ) @ V2rep1 ) @ V8rep3 ) ) @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27b @ A_27d @ A_27f @ A_27h ) @ V4abs2 ) @ V10abs4 ) ) @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27c @ A_27b @ A_27d ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27f @ A_27e @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V12f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27h @ A_27g @ A_27d ) @ V7abs3 ) @ V11rep4 ) @ V13g ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__sum_2ESUM__MAP__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27e ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27e @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27e ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27f ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27f @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27f ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27g ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27g @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27g ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9R4: $i] :
( ( mem @ V9R4 @ ( arr @ A_27d @ ( arr @ A_27d @ bool ) ) )
=> ! [V10abs4: $i] :
( ( mem @ V10abs4 @ ( arr @ A_27d @ A_27h ) )
=> ! [V11rep4: $i] :
( ( mem @ V11rep4 @ ( arr @ A_27h @ A_27d ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27d @ A_27h ) @ V9R4 ) @ V10abs4 ) @ V11rep4 ) )
=> ! [V12f1: $i] :
( ( mem @ V12f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V13f2: $i] :
( ( mem @ V13f2 @ ( arr @ A_27a @ A_27b ) )
=> ! [V14g1: $i] :
( ( mem @ V14g1 @ ( arr @ A_27c @ A_27d ) )
=> ! [V15g2: $i] :
( ( mem @ V15g2 @ ( arr @ A_27c @ A_27d ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V12f1 ) @ V13f2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27c @ A_27d ) @ V6R3 ) @ V9R4 ) @ V14g1 ) @ V15g2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ ( ty_2Esum_2Esum @ A_27a @ A_27c ) @ ( ty_2Esum_2Esum @ A_27b @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27a @ A_27c ) @ V0R1 ) @ V6R3 ) ) @ ( ap @ ( ap @ ( c_2Equotient__sum_2E_2B_2B_2B @ A_27b @ A_27d ) @ V3R2 ) @ V9R4 ) ) @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27c @ A_27b @ A_27d ) @ V12f1 ) @ V14g1 ) ) @ ( ap @ ( ap @ ( c_2Esum_2E_2B_2B @ A_27a @ A_27c @ A_27b @ A_27d ) @ V13f2 ) @ V15g2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------