ITP001 Axioms: ITP072^5.ax
%------------------------------------------------------------------------------
% File : ITP072^5 : 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_pair^2.ax [Gau20]
% : HL4072^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 21 ( 2 unt; 1 typ; 0 def)
% Number of atoms : 869 ( 11 equ; 0 cnn)
% Maximal formula atoms : 90 ( 41 avg)
% Number of connectives : 1855 ( 0 ~; 0 |; 4 &;1659 @)
% ( 3 <=>; 189 =>; 0 <=; 0 <~>)
% Maximal formula depth : 62 ( 31 avg;1659 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 4 ( 4 >; 0 *; 0 +; 0 <<)
% Number of symbols : 23 ( 22 usr; 21 con; 0-4 aty)
% Number of variables : 238 ( 4 ^ 234 !; 0 ?; 238 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Equotient__pair_2E_23_23_23,type,
c_2Equotient__pair_2E_23_23_23: del > del > del > del > $i ).
thf(mem_c_2Equotient__pair_2E_23_23_23,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27c @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27d @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) @ bool ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EPAIR__MAP__I,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ ( c_2Ecombin_2EI @ A_27b ) )
= ( c_2Ecombin_2EI @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(ax_thm_2Equotient__pair_2EPAIR__REL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27c @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27d @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0R1 ) @ V1R2 )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V2a: $i] :
( lam @ A_27b
@ ^ [V3b: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ bool )
@ ( lam @ A_27c
@ ^ [V4c: $i] :
( lam @ A_27d
@ ^ [V5d: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0R1 @ V2a ) @ V4c ) ) @ ( ap @ ( ap @ V1R2 @ V3b ) @ V5d ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EPAIR__REL__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27c @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27d @ bool ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ! [V4c: $i] :
( ( mem @ V4c @ A_27c )
=> ! [V5d: $i] :
( ( mem @ V5d @ A_27d )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0R1 ) @ V1R2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2a ) @ V3b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27d ) @ V4c ) @ V5d ) ) )
<=> ( ( p @ ( ap @ ( ap @ V0R1 @ V2a ) @ V4c ) )
& ( p @ ( ap @ ( ap @ V1R2 @ V3b ) @ V5d ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EPAIR__REL__EQ,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ ( c_2Emin_2E_3D @ A_27b ) )
= ( c_2Emin_2E_3D @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Equotient__pair_2EPAIR__REL__REFL,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 ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V2x ) @ V3y ) )
<=> ( ( ap @ V0R1 @ V2x )
= ( ap @ V0R1 @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V1R2 @ V4x ) @ V5y ) )
<=> ( ( ap @ V1R2 @ V4x )
= ( ap @ V1R2 @ V5y ) ) ) ) ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V6x ) @ V6x ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EPAIR__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_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EPAIR__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_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ V1abs1 ) @ V4abs2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27c @ A_27d @ A_27a @ A_27b ) @ V2rep1 ) @ V5rep2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EFST__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 ) )
=> ! [V6p: $i] :
( ( mem @ V6p @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) )
=> ( ( ap @ ( c_2Epair_2EFST @ A_27c @ A_27d ) @ V6p )
= ( ap @ V1abs1 @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27c @ A_27d @ A_27a @ A_27b ) @ V2rep1 ) @ V5rep2 ) @ V6p ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EFST__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 ) )
=> ! [V6p1: $i] :
( ( mem @ V6p1 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V7p2: $i] :
( ( mem @ V7p2 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6p1 ) @ V7p2 ) )
=> ( p @ ( ap @ ( ap @ V0R1 @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V6p1 ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V7p2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2ESND__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 ) )
=> ! [V6p: $i] :
( ( mem @ V6p @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) )
=> ( ( ap @ ( c_2Epair_2ESND @ A_27c @ A_27d ) @ V6p )
= ( ap @ V4abs2 @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27c @ A_27d @ A_27a @ A_27b ) @ V2rep1 ) @ V5rep2 ) @ V6p ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2ESND__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 ) )
=> ! [V6p1: $i] :
( ( mem @ V6p1 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V7p2: $i] :
( ( mem @ V7p2 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6p1 ) @ V7p2 ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V6p1 ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V7p2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2ECOMMA__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 )
=> ! [V7b: $i] :
( ( mem @ V7b @ A_27d )
=> ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27d ) @ V6a ) @ V7b )
= ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ V1abs1 ) @ V4abs2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( ap @ V2rep1 @ V6a ) ) @ ( ap @ V5rep2 @ V7b ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2ECOMMA__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_27b )
=> ! [V8b1: $i] :
( ( mem @ V8b1 @ A_27a )
=> ! [V9b2: $i] :
( ( mem @ V9b2 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ V0R1 @ V6a1 ) @ V8b1 ) )
& ( p @ ( ap @ ( ap @ V3R2 @ V7a2 ) @ V9b2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V6a1 ) @ V7a2 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V8b1 ) @ V9b2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2ECURRY__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27d ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27d @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27e ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27e @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27f ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27f @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) @ A_27f ) )
=> ! [V10a: $i] :
( ( mem @ V10a @ A_27d )
=> ! [V11b: $i] :
( ( mem @ V11b @ A_27e )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epair_2ECURRY @ A_27d @ A_27e @ A_27f ) @ V9f ) @ V10a ) @ V11b )
= ( ap @ V7abs3 @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27f @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27d @ A_27e ) @ V1abs1 ) @ V4abs2 ) ) @ V8rep3 ) @ V9f ) ) @ ( ap @ V2rep1 @ V10a ) ) @ ( ap @ V5rep2 @ V11b ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2ECURRY__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27d ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27d @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27e ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27e @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27f ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27f @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9f1: $i] :
( ( mem @ V9f1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
=> ! [V10f2: $i] :
( ( mem @ V10f2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6R3 ) @ V9f1 ) @ V10f2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ V0R1 ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c ) @ V3R2 ) @ V6R3 ) ) @ ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ V9f1 ) ) @ ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ V10f2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EUNCURRY__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27d ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27d @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27e ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27e @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27f ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27f @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27d @ ( arr @ A_27e @ A_27f ) ) )
=> ! [V10p: $i] :
( ( mem @ V10p @ ( ty_2Epair_2Eprod @ A_27d @ A_27e ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27d @ A_27e @ A_27f ) @ V9f ) @ V10p )
= ( ap @ V7abs3 @ ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( arr @ A_27e @ A_27f ) @ A_27d @ ( arr @ A_27b @ A_27c ) ) @ V1abs1 ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ A_27f @ A_27e @ A_27c ) @ V4abs2 ) @ V8rep3 ) ) @ V9f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27d @ A_27e @ A_27a @ A_27b ) @ V2rep1 ) @ V5rep2 ) @ V10p ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EUNCURRY__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27d ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27d @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27e ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27e @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27f ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27f @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9f1: $i] :
( ( mem @ V9f1 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V10f2: $i] :
( ( mem @ V10f2 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ V0R1 ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c ) @ V3R2 ) @ V6R3 ) ) @ V9f1 ) @ V10f2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6R3 ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V9f1 ) ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V10f2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pair_2EPAIR__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_2Epair_2E_23_23 @ A_27e @ A_27g @ A_27f @ A_27h ) @ V12f ) @ V13g )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ ( ty_2Epair_2Eprod @ A_27e @ A_27g ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27f @ A_27h ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27e @ A_27g @ A_27a @ A_27c ) @ V2rep1 ) @ V8rep3 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27b @ A_27d @ A_27f @ A_27h ) @ V4abs2 ) @ V10abs4 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ 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__pair_2EPAIR__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_2Epair_2Eprod @ A_27a @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27c @ A_27a @ A_27c ) @ V0R1 ) @ V6R3 ) ) @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27b @ A_27d @ A_27b @ A_27d ) @ V3R2 ) @ V9R4 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27c @ A_27b @ A_27d ) @ V12f1 ) @ V14g1 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27c @ A_27b @ A_27d ) @ V13f2 ) @ V15g2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------