TPTP Problem File: ITP009^5.p
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%------------------------------------------------------------------------------
% File : ITP009^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Equotient_2EFUN__REL__EQ__REL.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_2Equotient_2EFUN__REL__EQ__REL.p [Gau20]
% : HL404001^5.p [TPAP]
% Status : Theorem
% Rating : 1.00 v7.5.0
% Syntax : Number of formulae : 7111 ( 529 unt;1013 typ; 0 def)
% Number of atoms : 143499 (7357 equ; 0 cnn)
% Maximal formula atoms : 7693 ( 23 avg)
% Number of connectives : 347778 ( 681 ~; 416 |;17819 &;311507 @)
% (3330 <=>;14025 =>; 0 <=; 0 <~>)
% Maximal formula depth : 368 ( 12 avg)
% Number of types : 29 ( 27 usr)
% Number of type conns : 1139 (1139 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1005 (1002 usr; 235 con; 0-5 aty)
% Number of variables : 35039 (1735 ^;20635 !;12669 ?;35039 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
include('Axioms/ITP001/ITP002^5.ax').
include('Axioms/ITP001/ITP003^5.ax').
include('Axioms/ITP001/ITP004^5.ax').
include('Axioms/ITP001/ITP007^5.ax').
include('Axioms/ITP001/ITP006^5.ax').
include('Axioms/ITP001/ITP005^5.ax').
include('Axioms/ITP001/ITP008^5.ax').
include('Axioms/ITP001/ITP009^5.ax').
include('Axioms/ITP001/ITP010^5.ax').
include('Axioms/ITP001/ITP012^5.ax').
include('Axioms/ITP001/ITP011^5.ax').
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').
include('Axioms/ITP001/ITP036^5.ax').
include('Axioms/ITP001/ITP037^5.ax').
include('Axioms/ITP001/ITP039^5.ax').
include('Axioms/ITP001/ITP041^5.ax').
include('Axioms/ITP001/ITP042^5.ax').
include('Axioms/ITP001/ITP040^5.ax').
include('Axioms/ITP001/ITP044^5.ax').
include('Axioms/ITP001/ITP051^5.ax').
include('Axioms/ITP001/ITP045^5.ax').
include('Axioms/ITP001/ITP056^5.ax').
include('Axioms/ITP001/ITP046^5.ax').
include('Axioms/ITP001/ITP043^5.ax').
include('Axioms/ITP001/ITP052^5.ax').
include('Axioms/ITP001/ITP057^5.ax').
include('Axioms/ITP001/ITP048^5.ax').
include('Axioms/ITP001/ITP047^5.ax').
include('Axioms/ITP001/ITP055^5.ax').
include('Axioms/ITP001/ITP053^5.ax').
include('Axioms/ITP001/ITP054^5.ax').
include('Axioms/ITP001/ITP058^5.ax').
include('Axioms/ITP001/ITP049^5.ax').
%------------------------------------------------------------------------------
thf(tp_c_2Equotient_2E_2D_2D_3E,type,
c_2Equotient_2E_2D_2D_3E: del > del > del > del > $i ).
thf(mem_c_2Equotient_2E_2D_2D_3E,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27d ) @ ( arr @ ( arr @ A_27c @ A_27b ) @ ( arr @ A_27a @ A_27d ) ) ) ) ) ).
thf(tp_c_2Equotient_2E_3D_3D_3D_3E,type,
c_2Equotient_2E_3D_3D_3D_3E: del > del > $i ).
thf(mem_c_2Equotient_2E_3D_3D_3D_3E,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ) ).
thf(tp_c_2Equotient_2E_3F_21_21,type,
c_2Equotient_2E_3F_21_21: del > $i ).
thf(mem_c_2Equotient_2E_3F_21_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient_2E_3F_21_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(tp_c_2Equotient_2EEQUIV,type,
c_2Equotient_2EEQUIV: del > $i ).
thf(mem_c_2Equotient_2EEQUIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient_2EEQUIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Equotient_2EPARTIAL__EQUIV,type,
c_2Equotient_2EPARTIAL__EQUIV: del > $i ).
thf(mem_c_2Equotient_2EPARTIAL__EQUIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient_2EPARTIAL__EQUIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT: del > del > $i ).
thf(mem_c_2Equotient_2EQUOTIENT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ bool ) ) ) ) ).
thf(tp_c_2Equotient_2ERES__EXISTS__EQUIV,type,
c_2Equotient_2ERES__EXISTS__EQUIV: del > $i ).
thf(mem_c_2Equotient_2ERES__EXISTS__EQUIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Equotient_2Erespects,type,
c_2Equotient_2Erespects: del > del > $i ).
thf(mem_c_2Equotient_2Erespects,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2Erespects @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(ax_thm_2Equotient_2EEQUIV__def,axiom,
! [A_27a: del,V0E: $i] :
( ( mem @ V0E @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0E ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0E @ V1x ) @ V2y ) )
<=> ( ( ap @ V0E @ V1x )
= ( ap @ V0E @ V2y ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2EPARTIAL__EQUIV__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EPARTIAL__EQUIV @ A_27a ) @ V0R ) )
<=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V1x ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V2x ) )
& ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V3y ) )
& ( ( ap @ V0R @ V2x )
= ( ap @ V0R @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__IMP__PARTIAL__EQUIV,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Equotient_2EPARTIAL__EQUIV @ A_27a ) @ V0R ) ) ) ) ).
thf(ax_thm_2Equotient_2EQUOTIENT__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
<=> ( ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( ap @ V1abs @ ( ap @ V2rep @ V3a ) )
= V3a ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27b )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ V4a ) ) @ ( ap @ V2rep @ V4a ) ) ) )
& ! [V5r: $i] :
( ( mem @ V5r @ A_27a )
=> ! [V6s: $i] :
( ( mem @ V6s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V5r ) @ V6s ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V5r ) @ V5r ) )
& ( p @ ( ap @ ( ap @ V0R @ V6s ) @ V6s ) )
& ( ( ap @ V1abs @ V5r )
= ( ap @ V1abs @ V6s ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__ABS__REP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( ap @ V1abs @ ( ap @ V2rep @ V3a ) )
= V3a ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REP__REFL,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ V3a ) ) @ ( ap @ V2rep @ V3a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V4s ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V3r ) )
& ( p @ ( ap @ ( ap @ V0R @ V4s ) @ V4s ) )
& ( ( ap @ V1abs @ V3r )
= ( ap @ V1abs @ V4s ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL__ABS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V4s ) )
=> ( ( ap @ V1abs @ V3r )
= ( ap @ V1abs @ V4s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL__ABS__EQ,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V3r ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4s ) @ V4s ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V4s ) )
<=> ( ( ap @ V1abs @ V3r )
= ( ap @ V1abs @ V4s ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL__REP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ V3a ) ) @ ( ap @ V2rep @ V4b ) ) )
<=> ( V3a = V4b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REP__ABS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V3r ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ ( ap @ V1abs @ V3r ) ) ) @ V3r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EIDENTITY__EQUIV,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a ) ) ) ).
thf(conj_thm_2Equotient_2EIDENTITY__QUOTIENT,axiom,
! [A_27a: del] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ ( c_2Ecombin_2EI @ A_27a ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__REFL__SYM__TRANS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
<=> ( ( ap @ V0R @ V1x )
= ( ap @ V0R @ V2y ) ) ) ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V3x ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V0R @ V5y ) @ V4x ) ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ! [V8z: $i] :
( ( mem @ V8z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V6x ) @ V7y ) )
& ( p @ ( ap @ ( ap @ V0R @ V7y ) @ V8z ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V6x ) @ V8z ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__SYM,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__TRANS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V5z ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2EFUN__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ( ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1g )
= ( lam @ ( arr @ A_27c @ A_27b )
@ ^ [V2h: $i] :
( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V1g @ ( ap @ V2h @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__MAP__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27c @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1g ) @ V2h ) @ V3x )
= ( ap @ V1g @ ( ap @ V2h @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__MAP__I,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27a @ A_27b ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ ( c_2Ecombin_2EI @ A_27b ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Equotient_2EIN__FUN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ bool @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V0f ) @ V1g ) @ V2s ) ) )
<=> ( p @ ( ap @ V1g @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2EFUN__REL,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 ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V2f ) @ V3g ) )
<=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f @ V4x ) ) @ ( ap @ V3g @ V5y ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__REL__EQ,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ ( c_2Emin_2E_3D @ A_27b ) )
= ( c_2Emin_2E_3D @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__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 @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2Erespects__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Equotient_2Erespects @ A_27a @ A_27b )
= ( c_2Ecombin_2EW @ A_27a @ A_27b ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) @ V1x ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EIN__RESPECTS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS__THM,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 ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V2f ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS__MP,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 ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V2f ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V3x ) @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS__REP__ABS,axiom,
! [A_27a: del,A_27b: del,A_27c: 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 ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V4f ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V5x ) @ V5x ) ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ V4f @ ( ap @ V2rep1 @ ( ap @ V1abs1 @ V5x ) ) ) ) @ ( ap @ V4f @ V5x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2R3: $i] :
( ( mem @ V2R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ A_27c ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27b @ A_27c ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c ) @ V1R2 ) @ V2R3 ) ) @ V3f ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V4g ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27c ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27c ) @ V0R1 ) @ V2R3 ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V3f ) @ V4g ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2E_3F_21_21,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2E_3F_21_21 @ A_27a ) @ V0P ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V0P ) ) ) ) ).
thf(ax_thm_2Equotient_2ERES__EXISTS__EQUIV__DEF,axiom,
! [A_27a: del] :
( ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a )
= ( lam @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) )
@ ^ [V0R: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V1P: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V1P @ V2x ) ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V3x ) ) @ ( ap @ V1P @ V4y ) ) ) @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__EQUIV,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V0R ) @ V1m ) )
<=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V1m @ V2x ) ) ) )
& ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1m @ V3x ) ) @ ( ap @ V1m @ V4y ) ) ) @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__REL__EQ__REL,conjecture,
! [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 ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f ) @ V7g ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6f ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V7g ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V6f )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V7g ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------