ITP001 Axioms: ITP014^7.ax
%------------------------------------------------------------------------------
% File : ITP014^7 : 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 : pair.ax [Gau19]
% : HL4014^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 109 ( 42 unt; 35 typ; 0 def)
% Number of atoms : 159 ( 80 equ; 1 cnn)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 1175 ( 1 ~; 3 |; 26 &;1092 @)
% ( 28 <=>; 25 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg;1092 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 306 ( 306 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 34 usr; 1 con; 0-7 aty)
% Number of variables : 492 ( 41 ^ 377 !; 20 ?; 492 :)
% ( 54 !>; 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_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_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_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_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2EABS__prod,type,
c_2Epair_2EABS__prod:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ecombin_2EC,type,
c_2Ecombin_2EC:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > A_27b > A_27a > A_27c ) ).
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_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $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_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Epair_2ELEX,type,
c_2Epair_2ELEX:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Epair_2EPROD__ALL,type,
c_2Epair_2EPROD__ALL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Epair_2EREP__prod,type,
c_2Epair_2EREP__prod:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a > A_27b > $o ) ).
thf(c_2Epair_2ERPROD,type,
c_2Epair_2ERPROD:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Ecombin_2ES,type,
c_2Ecombin_2ES:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( A_27a > A_27b ) > A_27a > A_27c ) ).
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_2ESWAP,type,
c_2Epair_2ESWAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
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_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Epair_2Epair__CASE,type,
c_2Epair_2Epair__CASE:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > ( A_27b > A_27c > A_27a ) > A_27a ) ).
thf(c_2Erelation_2Ereflexive,type,
c_2Erelation_2Ereflexive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Esymmetric,type,
c_2Erelation_2Esymmetric:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Etotal,type,
c_2Erelation_2Etotal:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $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_2Epair_2Eprod__TY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
? [V0rep: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a > A_27b > $o] :
( c_2Ebool_2ETYPE__DEFINITION @ ( A_27a > A_27b > $o ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V1p: A_27a > A_27b > $o] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V2x: A_27a] :
( c_2Ebool_2E_3F @ A_27b
@ ^ [V3y: A_27b] :
( c_2Emin_2E_3D @ ( A_27a > A_27b > $o ) @ V1p
@ ^ [V4a: A_27a,V5b: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ V4a @ V2x ) @ ( c_2Emin_2E_3D @ A_27b @ V5b @ V3y ) ) ) ) )
@ V0rep ) ).
thf(thm_2Epair_2EABS__REP__prod,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2EABS__prod @ A_27a @ A_27b @ ( c_2Epair_2EREP__prod @ A_27a @ A_27b @ V0a ) )
= V0a )
& ! [V1r: A_27a > A_27b > $o] :
( ( ^ [V2p: A_27a > A_27b > $o] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V3x: A_27a] :
( c_2Ebool_2E_3F @ A_27b
@ ^ [V4y: A_27b] :
( c_2Emin_2E_3D @ ( A_27a > A_27b > $o ) @ V2p
@ ^ [V5a: A_27a,V6b: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ V5a @ V3x ) @ ( c_2Emin_2E_3D @ A_27b @ V6b @ V4y ) ) ) ) )
@ V1r )
<=> ( ( c_2Epair_2EREP__prod @ A_27a @ A_27b @ ( c_2Epair_2EABS__prod @ A_27a @ A_27b @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Epair_2ECOMMA__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Epair_2EABS__prod @ A_27a @ A_27b
@ ^ [V2a: A_27a,V3b: A_27b] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ V2a @ V0x ) @ ( c_2Emin_2E_3D @ A_27b @ V3b @ V1y ) ) ) ) ).
thf(thm_2Epair_2EPAIR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2E_2C @ A_27a @ A_27b @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V0x ) )
= V0x ) ).
thf(thm_2Epair_2ESWAP__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Epair_2Eprod @ A_27b @ A_27a] :
( ( c_2Epair_2ESWAP @ A_27a @ A_27b @ V0a )
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ ( c_2Epair_2ESND @ A_27b @ A_27a @ V0a ) @ ( c_2Epair_2EFST @ A_27b @ A_27a @ V0a ) ) ) ).
thf(thm_2Epair_2ECURRY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c,V1x: A_27a,V2y: A_27b] :
( ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c @ V0f @ V1x @ V2y )
= ( V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) ) ) ).
thf(thm_2Epair_2EUNCURRY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1v: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0f @ V1v )
= ( V0f @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V1v ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V1v ) ) ) ).
thf(thm_2Epair_2EPROD__ALL__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1Q: A_27b > $o,V2p: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b @ V0P @ V1Q @ V2p )
<=> ( ( V0P @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2p ) )
& ( V1Q @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2p ) ) ) ) ).
thf(thm_2Epair_2EPAIR__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27a > A_27c,V1g: A_27b > A_27d,V2p: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d @ V0f @ V1g @ V2p )
= ( c_2Epair_2E_2C @ A_27c @ A_27d @ ( V0f @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V2p ) ) @ ( V1g @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V2p ) ) ) ) ).
thf(thm_2Epair_2Epair__CASE__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p: tyop_2Epair_2Eprod @ A_27b @ A_27c,V1f: A_27b > A_27c > A_27a] :
( ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c @ V0p @ V1f )
= ( V1f @ ( c_2Epair_2EFST @ A_27b @ A_27c @ V0p ) @ ( c_2Epair_2ESND @ A_27b @ A_27c @ V0p ) ) ) ).
thf(thm_2Epair_2ELEX__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o] :
( ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R1 @ V1R2 )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o )
@ ^ [V2s: A_27a,V3t: A_27b] :
( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V4u: A_27a,V5v: A_27b] : ( c_2Ebool_2E_5C_2F @ ( V0R1 @ V2s @ V4u ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ V2s @ V4u ) @ ( V1R2 @ V3t @ V5v ) ) ) ) ) ) ).
thf(thm_2Epair_2ERPROD__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o] :
( ( c_2Epair_2ERPROD @ A_27a @ A_27b @ V0R1 @ V1R2 )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o )
@ ^ [V2s: A_27a,V3t: A_27b] :
( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V4u: A_27a,V5v: A_27b] : ( c_2Ebool_2E_2F_5C @ ( V0R1 @ V2s @ V4u ) @ ( V1R2 @ V3t @ V5v ) ) ) ) ) ).
thf(thm_2Epair_2EPAIR__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2b: A_27b,V3a: A_27a] :
( ( ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y )
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3a @ V2b ) )
<=> ( ( V1x = V3a )
& ( V0y = V2b ) ) ) ).
thf(thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b,V2a: A_27a,V3b: A_27b] :
( ( ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y )
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2a @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ).
thf(thm_2Epair_2EABS__PAIR__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
? [V1q: A_27a,V2r: A_27b] :
( V0x
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1q @ V2r ) ) ).
thf(thm_2Epair_2Epair__CASES,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
? [V1q: A_27a,V2r: A_27b] :
( V0x
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1q @ V2r ) ) ).
thf(thm_2Epair_2EFST,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( ( c_2Epair_2EFST @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y ) )
= V0x ) ).
thf(thm_2Epair_2ESND,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1y: A_27b] :
( ( c_2Epair_2ESND @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V1y ) )
= V1y ) ).
thf(thm_2Epair_2EPAIR__FST__SND__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1q: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( V0p = V1q )
<=> ( ( ( c_2Epair_2EFST @ A_27a @ A_27b @ V0p )
= ( c_2Epair_2EFST @ A_27a @ A_27b @ V1q ) )
& ( ( c_2Epair_2ESND @ A_27a @ A_27b @ V0p )
= ( c_2Epair_2ESND @ A_27a @ A_27b @ V1q ) ) ) ) ).
thf(thm_2Epair_2EUNCURRY__VAR,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1v: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0f @ V1v )
= ( V0f @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V1v ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V1v ) ) ) ).
thf(thm_2Epair_2EELIM__UNCURRY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c] :
( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0f )
= ( ^ [V1x: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V0f @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V1x ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V1x ) ) ) ) ).
thf(thm_2Epair_2EUNCURRY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c,V1x: A_27a,V2y: A_27b] :
( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0f @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) )
= ( V0f @ V1x @ V2y ) ) ).
thf(thm_2Epair_2ECURRY__UNCURRY__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c] :
( ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0f ) )
= V0f ) ).
thf(thm_2Epair_2EUNCURRY__CURRY__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c] :
( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c @ V0f ) )
= V0f ) ).
thf(thm_2Epair_2ECURRY__ONE__ONE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c,V1f: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c] :
( ( ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c @ V1f )
= ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c @ V0g ) )
<=> ( V1f = V0g ) ) ).
thf(thm_2Epair_2EUNCURRY__ONE__ONE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27a > A_27b > A_27c,V1f: A_27a > A_27b > A_27c] :
( ( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V1f )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0g ) )
<=> ( V1f = V0g ) ) ).
thf(thm_2Epair_2Epair__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c] :
? [V1fn: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c] :
! [V2x: A_27a,V3y: A_27b] :
( ( V1fn @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) )
= ( V0f @ V2x @ V3y ) ) ).
thf(thm_2Epair_2EUNCURRY__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_27: A_27a > A_27b > A_27c,V1f: A_27a > A_27b > A_27c,V2M_27: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3M: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( ( V3M = V2M_27 )
& ! [V4x: A_27a,V5y: A_27b] :
( ( V2M_27
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V4x @ V5y ) )
=> ( ( V1f @ V4x @ V5y )
= ( V0f_27 @ V4x @ V5y ) ) ) )
=> ( ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V1f @ V3M )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ V0f_27 @ V2M_27 ) ) ) ).
thf(thm_2Epair_2ELAMBDA__PROD,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c] :
( ( ^ [V1p: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V0P @ V1p ) )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c
@ ^ [V2p1: A_27a,V3p2: A_27b] : ( V0P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2p1 @ V3p2 ) ) ) ) ).
thf(thm_2Epair_2EEXISTS__PROD,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ? [V1p: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V0P @ V1p )
<=> ? [V2p__1: A_27a,V3p__2: A_27b] : ( V0P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2p__1 @ V3p__2 ) ) ) ).
thf(thm_2Epair_2EFORALL__PROD,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ! [V1p: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V0P @ V1p )
<=> ! [V2p__1: A_27a,V3p__2: A_27b] : ( V0P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2p__1 @ V3p__2 ) ) ) ).
thf(thm_2Epair_2Epair__induction,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ! [V1p__1: A_27a,V2p__2: A_27b] : ( V0P @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1p__1 @ V2p__2 ) )
=> ! [V3p: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V0P @ V3p ) ) ).
thf(thm_2Epair_2EPROD__ALL__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2Q: A_27b > $o,V3P: A_27a > $o] :
( ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b @ V3P @ V2Q @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y ) )
<=> ( ( V3P @ V1x )
& ( V2Q @ V0y ) ) ) ).
thf(thm_2Epair_2EPROD__ALL__MONO,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1Q_27: A_27b > $o,V2Q: A_27b > $o,V3P_27: A_27a > $o,V4P: A_27a > $o] :
( ( ! [V5x: A_27a] :
( ( V4P @ V5x )
=> ( V3P_27 @ V5x ) )
& ! [V6y: A_27b] :
( ( V2Q @ V6y )
=> ( V1Q_27 @ V6y ) ) )
=> ( ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b @ V4P @ V2Q @ V0p )
=> ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b @ V3P_27 @ V1Q_27 @ V0p ) ) ) ).
thf(thm_2Epair_2EPROD__ALL__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1p_27: tyop_2Epair_2Eprod @ A_27a @ A_27b,V2P: A_27a > $o,V3P_27: A_27a > $o,V4Q: A_27b > $o,V5Q_27: A_27b > $o] :
( ( ( V0p = V1p_27 )
& ! [V6x: A_27a,V7y: A_27b] :
( ( V1p_27
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V6x @ V7y ) )
=> ( ( V2P @ V6x )
= ( V3P_27 @ V6x ) ) )
& ! [V8x: A_27a,V9y: A_27b] :
( ( V1p_27
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V8x @ V9y ) )
=> ( ( V4Q @ V9y )
= ( V5Q_27 @ V9y ) ) ) )
=> ( ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b @ V2P @ V4Q @ V0p )
= ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b @ V3P_27 @ V5Q_27 @ V1p_27 ) ) ) ).
thf(thm_2Epair_2EELIM__PEXISTS,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ? [V1p: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V0P @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V1p ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V1p ) )
<=> ? [V2p1: A_27a,V3p2: A_27b] : ( V0P @ V2p1 @ V3p2 ) ) ).
thf(thm_2Epair_2EELIM__PFORALL,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ! [V1p: tyop_2Epair_2Eprod @ A_27a @ A_27b] : ( V0P @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V1p ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V1p ) )
<=> ! [V2p1: A_27a,V3p2: A_27b] : ( V0P @ V2p1 @ V3p2 ) ) ).
thf(thm_2Epair_2EPFORALL__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ! [V1x: A_27a,V2y: A_27b] : ( V0P @ V1x @ V2y )
<=> ( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V3x: A_27a,V4y: A_27b] : ( V0P @ V3x @ V4y ) ) ) ) ).
thf(thm_2Epair_2EPEXISTS__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ? [V1x: A_27a,V2y: A_27b] : ( V0P @ V1x @ V2y )
<=> ( c_2Ebool_2E_3F @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V3x: A_27a,V4y: A_27b] : ( V0P @ V3x @ V4y ) ) ) ) ).
thf(thm_2Epair_2EELIM__PEXISTS__EVAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ( c_2Ebool_2E_3F @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V1x: A_27a] : ( V0P @ V1x ) ) )
<=> ? [V2x: A_27a] : ( c_2Ebool_2E_3F @ A_27b @ ( V0P @ V2x ) ) ) ).
thf(thm_2Epair_2EELIM__PFORALL__EVAL,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > A_27b > $o] :
( ( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o
@ ^ [V1x: A_27a] : ( V0P @ V1x ) ) )
<=> ! [V2x: A_27a] : ( c_2Ebool_2E_21 @ A_27b @ ( V0P @ V2x ) ) ) ).
thf(thm_2Epair_2EPAIR__MAP__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27a > A_27b,V1g: A_27c > A_27d,V2x: A_27a,V3y: A_27c] :
( ( c_2Epair_2E_23_23 @ A_27a @ A_27c @ A_27b @ A_27d @ V0f @ V1g @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V2x @ V3y ) )
= ( c_2Epair_2E_2C @ A_27b @ A_27d @ ( V0f @ V2x ) @ ( V1g @ V3y ) ) ) ).
thf(thm_2Epair_2EFST__PAIR__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0p: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1f: A_27a > A_27c,V2g: A_27b > A_27d] :
( ( c_2Epair_2EFST @ A_27c @ A_27d @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d @ V1f @ V2g @ V0p ) )
= ( V1f @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0p ) ) ) ).
thf(thm_2Epair_2ESND__PAIR__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0p: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1f: A_27a > A_27c,V2g: A_27b > A_27d] :
( ( c_2Epair_2ESND @ A_27c @ A_27d @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d @ V1f @ V2g @ V0p ) )
= ( V2g @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V0p ) ) ) ).
thf(thm_2Epair_2ELET2__RAND,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0P: A_27c > A_27d,V1M: tyop_2Epair_2Eprod @ A_27a @ A_27b,V2N: A_27a > A_27b > A_27c] :
( ( V0P
@ ( c_2Ebool_2ELET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c
@ ^ [V3x: A_27a,V4y: A_27b] : ( V2N @ V3x @ V4y ) )
@ V1M ) )
= ( c_2Ebool_2ELET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27d
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27d
@ ^ [V5x: A_27a,V6y: A_27b] : ( V0P @ ( V2N @ V5x @ V6y ) ) )
@ V1M ) ) ).
thf(thm_2Epair_2ELET2__RATOR,axiom,
! [A_27a1: $tType,A_27a2: $tType,A_27b: $tType,A_27c: $tType,V0M: tyop_2Epair_2Eprod @ A_27a1 @ A_27a2,V1N: A_27a1 > A_27a2 > A_27b > A_27c,V2b: A_27b] :
( ( c_2Ebool_2ELET @ ( tyop_2Epair_2Eprod @ A_27a1 @ A_27a2 ) @ ( A_27b > A_27c )
@ ( c_2Epair_2EUNCURRY @ A_27a1 @ A_27a2 @ ( A_27b > A_27c )
@ ^ [V3x: A_27a1,V4y: A_27a2] : ( V1N @ V3x @ V4y ) )
@ V0M
@ V2b )
= ( c_2Ebool_2ELET @ ( tyop_2Epair_2Eprod @ A_27a1 @ A_27a2 ) @ A_27c
@ ( c_2Epair_2EUNCURRY @ A_27a1 @ A_27a2 @ A_27c
@ ^ [V5x: A_27a1,V6y: A_27a2] : ( V1N @ V5x @ V6y @ V2b ) )
@ V0M ) ) ).
thf(thm_2Epair_2Eo__UNCURRY__R,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g: A_27a > A_27b > A_27d,V1f: A_27d > A_27c] :
( ( c_2Ecombin_2Eo @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c @ A_27d @ V1f @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27d @ V0g ) )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ ( c_2Ecombin_2Eo @ A_27a @ ( A_27b > A_27c ) @ ( A_27b > A_27d ) @ ( c_2Ecombin_2Eo @ A_27b @ A_27c @ A_27d @ V1f ) @ V0g ) ) ) ).
thf(thm_2Epair_2EC__UNCURRY__L,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0x: A_27d,V1f: A_27a > A_27b > A_27d > A_27c] :
( ( c_2Ecombin_2EC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27d @ A_27c @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( A_27d > A_27c ) @ V1f ) @ V0x )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ ( c_2Ecombin_2EC @ A_27a @ A_27d @ ( A_27b > A_27c ) @ ( c_2Ecombin_2Eo @ A_27a @ ( A_27d > A_27b > A_27c ) @ ( A_27b > A_27d > A_27c ) @ ( c_2Ecombin_2EC @ A_27b @ A_27d @ A_27c ) @ V1f ) @ V0x ) ) ) ).
thf(thm_2Epair_2ES__UNCURRY__R,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g: A_27a > A_27b > A_27d,V1f: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27d > A_27c] :
( ( c_2Ecombin_2ES @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27d @ A_27c @ V1f @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27d @ V0g ) )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c @ ( c_2Ecombin_2ES @ A_27a @ ( A_27b > A_27d ) @ ( A_27b > A_27c ) @ ( c_2Ecombin_2Eo @ A_27a @ ( ( A_27b > A_27d ) > A_27b > A_27c ) @ ( A_27b > A_27d > A_27c ) @ ( c_2Ecombin_2ES @ A_27b @ A_27d @ A_27c ) @ ( c_2Ecombin_2Eo @ A_27a @ ( A_27b > A_27d > A_27c ) @ ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( c_2Ecombin_2Eo @ A_27b @ ( A_27d > A_27c ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1f ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) ) ) @ V0g ) ) ) ).
thf(thm_2Epair_2EFORALL__UNCURRY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b > $o] :
( ( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o @ V0f ) )
= ( c_2Ebool_2E_21 @ A_27a @ ( c_2Ecombin_2Eo @ A_27a @ $o @ ( A_27b > $o ) @ ( c_2Ebool_2E_21 @ A_27b ) @ V0f ) ) ) ).
thf(thm_2Epair_2EPAIR__FUN__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0P: ( A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) ) > $o] :
( ( c_2Ebool_2E_3F_21 @ ( A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) )
@ ^ [V1f: A_27a > ( tyop_2Epair_2Eprod @ A_27b @ A_27c )] : ( V0P @ V1f ) )
= ( c_2Ebool_2E_3F_21 @ ( tyop_2Epair_2Eprod @ ( A_27a > A_27b ) @ ( A_27a > A_27c ) )
@ ^ [V2p: tyop_2Epair_2Eprod @ ( A_27a > A_27b ) @ ( A_27a > A_27c )] :
( V0P
@ ^ [V3a: A_27a] : ( c_2Epair_2E_2C @ A_27b @ A_27c @ ( c_2Epair_2EFST @ ( A_27a > A_27b ) @ ( A_27a > A_27c ) @ V2p @ V3a ) @ ( c_2Epair_2ESND @ ( A_27a > A_27b ) @ ( A_27a > A_27c ) @ V2p @ V3a ) ) ) ) ) ).
thf(thm_2Epair_2Epair__case__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0y: A_27c,V1x: A_27b,V2f: A_27b > A_27c > A_27a] :
( ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c @ ( c_2Epair_2E_2C @ A_27b @ A_27c @ V1x @ V0y ) @ V2f )
= ( V2f @ V1x @ V0y ) ) ).
thf(thm_2Epair_2Epair__case__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0y: A_27c,V1x: A_27b,V2f: A_27b > A_27c > A_27a] :
( ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c @ ( c_2Epair_2E_2C @ A_27b @ A_27c @ V1x @ V0y ) @ V2f )
= ( V2f @ V1x @ V0y ) ) ).
thf(thm_2Epair_2Epair__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_27: A_27b > A_27c > A_27a,V1M: tyop_2Epair_2Eprod @ A_27b @ A_27c,V2M_27: tyop_2Epair_2Eprod @ A_27b @ A_27c,V3f: A_27b > A_27c > A_27a] :
( ( ( V1M = V2M_27 )
& ! [V4x: A_27b,V5y: A_27c] :
( ( V2M_27
= ( c_2Epair_2E_2C @ A_27b @ A_27c @ V4x @ V5y ) )
=> ( ( V3f @ V4x @ V5y )
= ( V0f_27 @ V4x @ V5y ) ) ) )
=> ( ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c @ V1M @ V3f )
= ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c @ V2M_27 @ V0f_27 ) ) ) ).
thf(thm_2Epair_2Epair__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v: A_27a,V1p: tyop_2Epair_2Eprod @ A_27b @ A_27c,V2f: A_27b > A_27c > A_27a] :
( ( ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c @ V1p @ V2f )
= V0v )
<=> ? [V3x: A_27b,V4y: A_27c] :
( ( V1p
= ( c_2Epair_2E_2C @ A_27b @ A_27c @ V3x @ V4y ) )
& ( ( V2f @ V3x @ V4y )
= V0v ) ) ) ).
thf(thm_2Epair_2Edatatype__pair,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0pair: ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) > A_27c] : ( c_2Ebool_2EDATATYPE @ A_27c @ ( V0pair @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) ) ) ).
thf(thm_2Epair_2ELEX__DEF__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0d: A_27b,V1c: A_27a,V2b: A_27b,V3a: A_27a,V4R2: A_27b > A_27b > $o,V5R1: A_27a > A_27a > $o] :
( ( c_2Epair_2ELEX @ A_27a @ A_27b @ V5R1 @ V4R2 @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3a @ V2b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1c @ V0d ) )
<=> ( ( V5R1 @ V3a @ V1c )
| ( ( V3a = V1c )
& ( V4R2 @ V2b @ V0d ) ) ) ) ).
thf(thm_2Epair_2ELEX__MONO,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1x: tyop_2Epair_2Eprod @ A_27a @ A_27b,V2R4: A_27b > A_27b > $o,V3R3: A_27b > A_27b > $o,V4R2: A_27a > A_27a > $o,V5R1: A_27a > A_27a > $o] :
( ( ! [V6x: A_27a,V7y: A_27a] :
( ( V5R1 @ V6x @ V7y )
=> ( V4R2 @ V6x @ V7y ) )
& ! [V8x: A_27b,V9y: A_27b] :
( ( V3R3 @ V8x @ V9y )
=> ( V2R4 @ V8x @ V9y ) ) )
=> ( ( c_2Epair_2ELEX @ A_27a @ A_27b @ V5R1 @ V3R3 @ V1x @ V0y )
=> ( c_2Epair_2ELEX @ A_27a @ A_27b @ V4R2 @ V2R4 @ V1x @ V0y ) ) ) ).
thf(thm_2Epair_2EWF__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1Q: A_27b > A_27b > $o] :
( ( ( c_2Erelation_2EWF @ A_27a @ V0R )
& ( c_2Erelation_2EWF @ A_27b @ V1Q ) )
=> ( c_2Erelation_2EWF @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R @ V1Q ) ) ) ).
thf(thm_2Epair_2EWF__RPROD,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1Q: A_27b > A_27b > $o] :
( ( ( c_2Erelation_2EWF @ A_27a @ V0R )
& ( c_2Erelation_2EWF @ A_27b @ V1Q ) )
=> ( c_2Erelation_2EWF @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ERPROD @ A_27a @ A_27b @ V0R @ V1Q ) ) ) ).
thf(thm_2Epair_2Etotal__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2: A_27b > A_27b > $o,V1R1: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Etotal @ A_27a @ V1R1 )
& ( c_2Erelation_2Etotal @ A_27b @ V0R2 ) )
=> ( c_2Erelation_2Etotal @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V1R1 @ V0R2 ) ) ) ).
thf(thm_2Epair_2Etransitive__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2: A_27b > A_27b > $o,V1R1: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Etransitive @ A_27a @ V1R1 )
& ( c_2Erelation_2Etransitive @ A_27b @ V0R2 ) )
=> ( c_2Erelation_2Etransitive @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V1R1 @ V0R2 ) ) ) ).
thf(thm_2Epair_2Ereflexive__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2: A_27b > A_27b > $o,V1R1: A_27a > A_27a > $o] :
( ( c_2Erelation_2Ereflexive @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V1R1 @ V0R2 ) )
<=> ( ( c_2Erelation_2Ereflexive @ A_27a @ V1R1 )
| ( c_2Erelation_2Ereflexive @ A_27b @ V0R2 ) ) ) ).
thf(thm_2Epair_2Esymmetric__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2: A_27b > A_27b > $o,V1R1: A_27a > A_27a > $o] :
( ( ( c_2Erelation_2Esymmetric @ A_27a @ V1R1 )
& ( c_2Erelation_2Esymmetric @ A_27b @ V0R2 ) )
=> ( c_2Erelation_2Esymmetric @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2ELEX @ A_27a @ A_27b @ V1R1 @ V0R2 ) ) ) ).
thf(thm_2Epair_2ELEX__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2v1: tyop_2Epair_2Eprod @ A_27a @ A_27b,V3v2: tyop_2Epair_2Eprod @ A_27a @ A_27b,V4R1_27: A_27a > A_27a > $o,V5R2_27: A_27b > A_27b > $o,V6v1_27: tyop_2Epair_2Eprod @ A_27a @ A_27b,V7v2_27: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( ( V2v1 = V6v1_27 )
& ( V3v2 = V7v2_27 )
& ! [V8a: A_27a,V9b: A_27b,V10c: A_27a,V11d: A_27b] :
( ( ( V6v1_27
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V8a @ V9b ) )
& ( V7v2_27
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V10c @ V11d ) ) )
=> ( ( V0R1 @ V8a @ V10c )
= ( V4R1_27 @ V8a @ V10c ) ) )
& ! [V12a: A_27a,V13b: A_27b,V14c: A_27a,V15d: A_27b] :
( ( ( V6v1_27
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V12a @ V13b ) )
& ( V7v2_27
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V14c @ V15d ) )
& ( V12a = V14c ) )
=> ( ( V1R2 @ V13b @ V15d )
= ( V5R2_27 @ V13b @ V15d ) ) ) )
=> ( ( c_2Epair_2ELEX @ A_27a @ A_27b @ V0R1 @ V1R2 @ V2v1 @ V3v2 )
= ( c_2Epair_2ELEX @ A_27a @ A_27b @ V4R1_27 @ V5R2_27 @ V6v1_27 @ V7v2_27 ) ) ) ).
thf(thm_2Epair_2EFST__EQ__EQUIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1p: tyop_2Epair_2Eprod @ A_27a @ A_27b] :
( ( ( c_2Epair_2EFST @ A_27a @ A_27b @ V1p )
= V0x )
<=> ? [V2y: A_27b] :
( V1p
= ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V2y ) ) ) ).
thf(thm_2Epair_2ESND__EQ__EQUIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27a,V1p: tyop_2Epair_2Eprod @ A_27b @ A_27a] :
( ( ( c_2Epair_2ESND @ A_27b @ A_27a @ V1p )
= V0y )
<=> ? [V2x: A_27b] :
( V1p
= ( c_2Epair_2E_2C @ A_27b @ A_27a @ V2x @ V0y ) ) ) ).
%------------------------------------------------------------------------------