ITP001 Axioms: ITP072^7.ax
%------------------------------------------------------------------------------
% File : ITP072^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : quotient_pair.ax [Gau19]
% : HL4072^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 48 ( 9 unt; 22 typ; 0 def)
% Number of atoms : 84 ( 12 equ; 1 cnn)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 732 ( 1 ~; 1 |; 5 &; 671 @)
% ( 10 <=>; 44 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 17 avg; 671 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 244 ( 244 >; 0 *; 0 +; 0 <<)
% Number of symbols : 23 ( 21 usr; 1 con; 0-8 aty)
% Number of variables : 283 ( 4 ^ 245 !; 1 ?; 283 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_23_23,type,
c_2Epair_2E_23_23:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) ) ).
thf(c_2Equotient__pair_2E_23_23_23,type,
c_2Equotient__pair_2E_23_23_23:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c > $o ) > ( A_27b > A_27d > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Equotient_2E_2D_2D_3E,type,
c_2Equotient_2E_2D_2D_3E:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( A_27c > A_27b ) > A_27a > A_27d ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Equotient_2E_3D_3D_3D_3E,type,
c_2Equotient_2E_3D_3D_3D_3E:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( A_27a > A_27b ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2ECURRY,type,
c_2Epair_2ECURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) > A_27a > A_27b > A_27c ) ).
thf(c_2Equotient_2EEQUIV,type,
c_2Equotient_2EEQUIV:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Equotient__pair_2EPAIR__REL,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27c > $o,V1R2: A_27b > A_27d > $o] :
( ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27c @ A_27d @ V0R1 @ V1R2 )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) > $o )
@ ^ [V2a: A_27a,V3b: A_27b] :
( c_2Epair_2EUNCURRY @ A_27c @ A_27d @ $o
@ ^ [V4c: A_27c,V5d: A_27d] : ( c_2Ebool_2E_2F_5C @ ( V0R1 @ V2a @ V4c ) @ ( V1R2 @ V3b @ V5d ) ) ) ) ) ).
thf(thm_2Equotient__pair_2EPAIR__MAP__I,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( 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 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(thm_2Equotient__pair_2EPAIR__REL__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27c > $o,V1R2: A_27b > A_27d > $o,V2a: A_27a,V3b: A_27b,V4c: A_27c,V5d: A_27d] :
( ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27c @ A_27d @ V0R1 @ V1R2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2a @ V3b ) @ ( c_2Epair_2E_2C @ A_27c @ A_27d @ V4c @ V5d ) )
<=> ( ( V0R1 @ V2a @ V4c )
& ( V1R2 @ V3b @ V5d ) ) ) ).
thf(thm_2Equotient__pair_2EPAIR__REL__EQ,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( 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 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(thm_2Equotient__pair_2EPAIR__REL__REFL,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( V0R1 @ V2x @ V3y )
<=> ( ( V0R1 @ V2x )
= ( V0R1 @ V3y ) ) )
& ! [V4x: A_27b,V5y: A_27b] :
( ( V1R2 @ V4x @ V5y )
<=> ( ( V1R2 @ V4x )
= ( V1R2 @ V5y ) ) ) )
=> ! [V6x: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b @ V0R1 @ V1R2 @ V6x @ V6x ) ) ).
thf(thm_2Equotient__pair_2EPAIR__EQUIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V0R1 )
=> ( ( c_2Equotient_2EEQUIV @ A_27b @ V1R2 )
=> ( c_2Equotient_2EEQUIV @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b @ V0R1 @ V1R2 ) ) ) ) ).
thf(thm_2Equotient__pair_2EPAIR__QUOTIENT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ( c_2Equotient_2EQUOTIENT @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27c @ A_27d ) @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d @ V1abs1 @ V4abs2 ) @ ( c_2Epair_2E_23_23 @ A_27c @ A_27d @ A_27a @ A_27b @ V2rep1 @ V5rep2 ) ) ) ) ).
thf(thm_2Equotient__pair_2EFST__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6p: tyop_2Epair_2Eprod @ A_27c @ A_27d] :
( ( c_2Epair_2EFST @ A_27c @ A_27d @ V6p )
= ( V1abs1 @ ( c_2Epair_2EFST @ A_27a @ A_27b @ ( c_2Epair_2E_23_23 @ A_27c @ A_27d @ A_27a @ A_27b @ V2rep1 @ V5rep2 @ V6p ) ) ) ) ) ) ).
thf(thm_2Equotient__pair_2EFST__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6p1: tyop_2Epair_2Eprod @ A_27a @ A_27b,V7p2: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6p1 @ V7p2 )
=> ( V0R1 @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V6p1 ) @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V7p2 ) ) ) ) ) ).
thf(thm_2Equotient__pair_2ESND__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6p: tyop_2Epair_2Eprod @ A_27c @ A_27d] :
( ( c_2Epair_2ESND @ A_27c @ A_27d @ V6p )
= ( V4abs2 @ ( c_2Epair_2ESND @ A_27a @ A_27b @ ( c_2Epair_2E_23_23 @ A_27c @ A_27d @ A_27a @ A_27b @ V2rep1 @ V5rep2 @ V6p ) ) ) ) ) ) ).
thf(thm_2Equotient__pair_2ESND__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6p1: tyop_2Epair_2Eprod @ A_27a @ A_27b,V7p2: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6p1 @ V7p2 )
=> ( V3R2 @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V6p1 ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V7p2 ) ) ) ) ) ).
thf(thm_2Equotient__pair_2ECOMMA__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6a: A_27c,V7b: A_27d] :
( ( c_2Epair_2E_2C @ A_27c @ A_27d @ V6a @ V7b )
= ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d @ V1abs1 @ V4abs2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ ( V2rep1 @ V6a ) @ ( V5rep2 @ V7b ) ) ) ) ) ) ).
thf(thm_2Equotient__pair_2ECOMMA__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6a1: A_27a,V7a2: A_27b,V8b1: A_27a,V9b2: A_27b] :
( ( ( V0R1 @ V6a1 @ V8b1 )
& ( V3R2 @ V7a2 @ V9b2 ) )
=> ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b @ V0R1 @ V3R2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V6a1 @ V7a2 ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V8b1 @ V9b2 ) ) ) ) ) ).
thf(thm_2Equotient__pair_2ECURRY__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27d,V2rep1: A_27d > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27e,V5rep2: A_27e > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27f,V8rep3: A_27f > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9f: ( tyop_2Epair_2Eprod @ A_27d @ A_27e ) > A_27f,V10a: A_27d,V11b: A_27e] :
( ( c_2Epair_2ECURRY @ A_27d @ A_27e @ A_27f @ V9f @ V10a @ V11b )
= ( V7abs3 @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c @ ( c_2Equotient_2E_2D_2D_3E @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27f @ ( tyop_2Epair_2Eprod @ A_27d @ A_27e ) @ A_27c @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27d @ A_27e @ V1abs1 @ V4abs2 ) @ V8rep3 @ V9f ) @ ( V2rep1 @ V10a ) @ ( V5rep2 @ V11b ) ) ) ) ) ) ) ).
thf(thm_2Equotient__pair_2ECURRY__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27d,V2rep1: A_27d > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27e,V5rep2: A_27e > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27f,V8rep3: A_27f > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9f1: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c,V10f2: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V6R3 @ V9f1 @ V10f2 )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( A_27b > A_27c ) @ V0R1 @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c @ V3R2 @ V6R3 ) @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c @ V9f1 ) @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c @ V10f2 ) ) ) ) ) ) ).
thf(thm_2Equotient__pair_2EUNCURRY__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27d,V2rep1: A_27d > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27e,V5rep2: A_27e > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27f,V8rep3: A_27f > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9f: A_27d > A_27e > A_27f,V10p: tyop_2Epair_2Eprod @ A_27d @ A_27e] :
( ( c_2Epair_2EUNCURRY @ A_27d @ A_27e @ A_27f @ V9f @ V10p )
= ( V7abs3 @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( A_27e > A_27f ) @ A_27d @ ( A_27b > A_27c ) @ V1abs1 @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ A_27f @ A_27e @ A_27c @ V4abs2 @ V8rep3 ) @ V9f ) @ ( c_2Epair_2E_23_23 @ A_27d @ A_27e @ A_27a @ A_27b @ V2rep1 @ V5rep2 @ V10p ) ) ) ) ) ) ) ).
thf(thm_2Equotient__pair_2EUNCURRY__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27d,V2rep1: A_27d > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27e,V5rep2: A_27e > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27f,V8rep3: A_27f > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9f1: A_27a > A_27b > A_27c,V10f2: A_27a > A_27b > A_27c] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( A_27b > A_27c ) @ V0R1 @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c @ V3R2 @ V6R3 ) @ V9f1 @ V10f2 )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27b @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V6R3 @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V9f1 ) @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V10f2 ) ) ) ) ) ) ).
thf(thm_2Equotient__pair_2EPAIR__MAP__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27e,V2rep1: A_27e > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27e @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27f,V5rep2: A_27f > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27f @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27g,V8rep3: A_27g > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27g @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9R4: A_27d > A_27d > $o,V10abs4: A_27d > A_27h,V11rep4: A_27h > A_27d] :
( ( c_2Equotient_2EQUOTIENT @ A_27d @ A_27h @ V9R4 @ V10abs4 @ V11rep4 )
=> ! [V12f: A_27e > A_27f,V13g: A_27g > A_27h] :
( ( c_2Epair_2E_23_23 @ A_27e @ A_27g @ A_27f @ A_27h @ V12f @ V13g )
= ( c_2Equotient_2E_2D_2D_3E @ ( tyop_2Epair_2Eprod @ A_27e @ A_27g ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27f @ A_27h ) @ ( c_2Epair_2E_23_23 @ A_27e @ A_27g @ A_27a @ A_27c @ V2rep1 @ V8rep3 ) @ ( c_2Epair_2E_23_23 @ A_27b @ A_27d @ A_27f @ A_27h @ V4abs2 @ V10abs4 ) @ ( c_2Epair_2E_23_23 @ A_27a @ A_27c @ A_27b @ A_27d @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27f @ A_27e @ A_27b @ V1abs1 @ V5rep2 @ V12f ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27h @ A_27g @ A_27d @ V7abs3 @ V11rep4 @ V13g ) ) ) ) ) ) ) ) ).
thf(thm_2Equotient__pair_2EPAIR__MAP__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,A_27g: $tType,A_27h: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27e,V2rep1: A_27e > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27e @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27f,V5rep2: A_27f > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27f @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27g,V8rep3: A_27g > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27g @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9R4: A_27d > A_27d > $o,V10abs4: A_27d > A_27h,V11rep4: A_27h > A_27d] :
( ( c_2Equotient_2EQUOTIENT @ A_27d @ A_27h @ V9R4 @ V10abs4 @ V11rep4 )
=> ! [V12f1: A_27a > A_27b,V13f2: A_27a > A_27b,V14g1: A_27c > A_27d,V15g2: A_27c > A_27d] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V12f1 @ V13f2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27c @ A_27d @ V6R3 @ V9R4 @ V14g1 @ V15g2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ A_27c @ A_27a @ A_27c @ V0R1 @ V6R3 ) @ ( c_2Equotient__pair_2E_23_23_23 @ A_27b @ A_27d @ A_27b @ A_27d @ V3R2 @ V9R4 ) @ ( c_2Epair_2E_23_23 @ A_27a @ A_27c @ A_27b @ A_27d @ V12f1 @ V14g1 ) @ ( c_2Epair_2E_23_23 @ A_27a @ A_27c @ A_27b @ A_27d @ V13f2 @ V15g2 ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------