ITP001 Axioms: ITP014^5.ax
%------------------------------------------------------------------------------
% File : ITP014^5 : 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 : pair^2.ax [Gau20]
% : HL4014^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 94 ( 0 unt; 14 typ; 0 def)
% Number of atoms : 1254 ( 71 equ; 0 cnn)
% Maximal formula atoms : 57 ( 13 avg)
% Number of connectives : 2943 ( 0 ~; 2 |; 44 &;2653 @)
% ( 29 <=>; 215 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 15 avg;2653 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 33 ( 33 >; 0 *; 0 +; 0 <<)
% Number of symbols : 39 ( 38 usr; 24 con; 0-4 aty)
% Number of variables : 453 ( 41 ^ 393 !; 19 ?; 453 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Epair_2Eprod,type,
ty_2Epair_2Eprod: del > del > del ).
thf(tp_c_2Epair_2E_23_23,type,
c_2Epair_2E_23_23: del > del > del > del > $i ).
thf(mem_c_2Epair_2E_23_23,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27d ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) ) ) ) ) ).
thf(tp_c_2Epair_2E_2C,type,
c_2Epair_2E_2C: del > del > $i ).
thf(mem_c_2Epair_2E_2C,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Epair_2EABS__prod,type,
c_2Epair_2EABS__prod: del > del > $i ).
thf(mem_c_2Epair_2EABS__prod,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2EABS__prod @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Epair_2ECURRY,type,
c_2Epair_2ECURRY: del > del > del > $i ).
thf(mem_c_2Epair_2ECURRY,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) ) ) ).
thf(tp_c_2Epair_2EFST,type,
c_2Epair_2EFST: del > del > $i ).
thf(mem_c_2Epair_2EFST,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27a ) ) ).
thf(tp_c_2Epair_2ELEX,type,
c_2Epair_2ELEX: del > del > $i ).
thf(mem_c_2Epair_2ELEX,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ) ) ).
thf(tp_c_2Epair_2EPROD__ALL,type,
c_2Epair_2EPROD__ALL: del > del > $i ).
thf(mem_c_2Epair_2EPROD__ALL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Epair_2EREP__prod,type,
c_2Epair_2EREP__prod: del > del > $i ).
thf(mem_c_2Epair_2EREP__prod,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2EREP__prod @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ).
thf(tp_c_2Epair_2ERPROD,type,
c_2Epair_2ERPROD: del > del > $i ).
thf(mem_c_2Epair_2ERPROD,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2ERPROD @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) ) ) ) ) ).
thf(tp_c_2Epair_2ESND,type,
c_2Epair_2ESND: del > del > $i ).
thf(mem_c_2Epair_2ESND,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27b ) ) ).
thf(tp_c_2Epair_2ESWAP,type,
c_2Epair_2ESWAP: del > del > $i ).
thf(mem_c_2Epair_2ESWAP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Epair_2ESWAP @ A_27a @ A_27b ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY: del > del > del > $i ).
thf(mem_c_2Epair_2EUNCURRY,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) ) ) ).
thf(tp_c_2Epair_2Epair__CASE,type,
c_2Epair_2Epair__CASE: del > del > del > $i ).
thf(mem_c_2Epair_2Epair__CASE,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ A_27a ) ) ) ).
thf(ax_thm_2Epair_2Eprod__TY__DEF,axiom,
! [A_27a: del,A_27b: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( lam @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) )
@ ^ [V1p: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V3y: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) @ V1p )
@ ( lam @ A_27a
@ ^ [V4a: $i] :
( lam @ A_27b
@ ^ [V5b: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V4a ) @ V2x ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V5b ) @ V3y ) ) ) ) ) ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Epair_2EABS__REP__prod,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Epair_2EABS__prod @ A_27a @ A_27b ) @ ( ap @ ( c_2Epair_2EREP__prod @ A_27a @ A_27b ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p
@ ( ap
@ ( lam @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) )
@ ^ [V2p: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V4y: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) @ V2p )
@ ( lam @ A_27a
@ ^ [V5a: $i] :
( lam @ A_27b
@ ^ [V6b: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V5a ) @ V3x ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V6b ) @ V4y ) ) ) ) ) ) ) ) ) )
@ V1r ) )
<=> ( ( ap @ ( c_2Epair_2EREP__prod @ A_27a @ A_27b ) @ ( ap @ ( c_2Epair_2EABS__prod @ A_27a @ A_27b ) @ V1r ) )
= V1r ) ) ) ) ).
thf(ax_thm_2Epair_2ECOMMA__DEF,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y )
= ( ap @ ( c_2Epair_2EABS__prod @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V2a: $i] :
( lam @ A_27b
@ ^ [V3b: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2a ) @ V0x ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ V3b ) @ V1y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPAIR__EQ,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2a ) @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2a ) @ V3b ) )
<=> ( ( V0x = V2a )
& ( V1y = V3b ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EABS__PAIR__THM,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ? [V1q: $i] :
( ( mem @ V1q @ A_27a )
& ? [V2r: $i] :
( ( mem @ V2r @ A_27b )
& ( V0x
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1q ) @ V2r ) ) ) ) ) ).
thf(conj_thm_2Epair_2Epair__CASES,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ? [V1q: $i] :
( ( mem @ V1q @ A_27a )
& ? [V2r: $i] :
( ( mem @ V2r @ A_27b )
& ( V0x
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1q ) @ V2r ) ) ) ) ) ).
thf(ax_thm_2Epair_2EPAIR,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0x ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0x ) )
= V0x ) ) ).
thf(conj_thm_2Epair_2EFST,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
= V0x ) ) ) ).
thf(conj_thm_2Epair_2ESND,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
= V1y ) ) ) ).
thf(conj_thm_2Epair_2EPAIR__FST__SND__EQ,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1q: $i] :
( ( mem @ V1q @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( V0p = V1q )
<=> ( ( ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0p )
= ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1q ) )
& ( ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0p )
= ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V1q ) ) ) ) ) ) ).
thf(ax_thm_2Epair_2ESWAP__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) )
=> ( ( ap @ ( c_2Epair_2ESWAP @ A_27a @ A_27b ) @ V0a )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ ( ap @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ V0a ) ) @ ( ap @ ( c_2Epair_2EFST @ A_27b @ A_27a ) @ V0a ) ) ) ) ).
thf(ax_thm_2Epair_2ECURRY__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1x ) @ V2y )
= ( ap @ V0f @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(ax_thm_2Epair_2EUNCURRY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1v )
= ( ap @ ( ap @ V0f @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1v ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V1v ) ) ) ) ) ).
thf(conj_thm_2Epair_2EUNCURRY__VAR,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1v )
= ( ap @ ( ap @ V0f @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1v ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V1v ) ) ) ) ) ).
thf(conj_thm_2Epair_2EELIM__UNCURRY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V0f )
= ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V1x: $i] : ( ap @ ( ap @ V0f @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1x ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Epair_2EUNCURRY__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V0f ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) )
= ( ap @ ( ap @ V0f @ V1x ) @ V2y ) ) ) ) ) ).
thf(conj_thm_2Epair_2ECURRY__UNCURRY__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V0f ) )
= V0f ) ) ).
thf(conj_thm_2Epair_2EUNCURRY__CURRY__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
=> ( ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ V0f ) )
= V0f ) ) ).
thf(conj_thm_2Epair_2ECURRY__ONE__ONE__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
=> ( ( ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ V0f )
= ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27b @ A_27c ) @ V1g ) )
<=> ( V0f = V1g ) ) ) ) ).
thf(conj_thm_2Epair_2EUNCURRY__ONE__ONE__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V0f )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V1g ) )
<=> ( V0f = V1g ) ) ) ) ).
thf(conj_thm_2Epair_2Epair__Axiom,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ? [V1fn: $i] :
( ( mem @ V1fn @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ap @ V1fn @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) )
= ( ap @ ( ap @ V0f @ V2x ) @ V3y ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EUNCURRY__CONG,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f_27: $i] :
( ( mem @ V0f_27 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V2M_27: $i] :
( ( mem @ V2M_27 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V3M: $i] :
( ( mem @ V3M @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ( V3M = V2M_27 )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27b )
=> ( ( V2M_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V4x ) @ V5y ) )
=> ( ( ap @ ( ap @ V1f @ V4x ) @ V5y )
= ( ap @ ( ap @ V0f_27 @ V4x ) @ V5y ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V1f ) @ V3M )
= ( ap @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ V0f_27 ) @ V2M_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ELAMBDA__PROD,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c ) )
=> ( ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27b )
@ ^ [V1p: $i] : ( ap @ V0P @ V1p ) )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c )
@ ( lam @ A_27a
@ ^ [V2p1: $i] :
( lam @ A_27b
@ ^ [V3p2: $i] : ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2p1 ) @ V3p2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EEXISTS__PROD,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ? [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
& ( p @ ( ap @ V0P @ V1p ) ) )
<=> ? [V2p__1: $i] :
( ( mem @ V2p__1 @ A_27a )
& ? [V3p__2: $i] :
( ( mem @ V3p__2 @ A_27b )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2p__1 ) @ V3p__2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EFORALL__PROD,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V1p ) ) )
<=> ! [V2p__1: $i] :
( ( mem @ V2p__1 @ A_27a )
=> ! [V3p__2: $i] :
( ( mem @ V3p__2 @ A_27b )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2p__1 ) @ V3p__2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2Epair__induction,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
=> ( ! [V1p__1: $i] :
( ( mem @ V1p__1 @ A_27a )
=> ! [V2p__2: $i] :
( ( mem @ V2p__2 @ A_27b )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1p__1 ) @ V2p__2 ) ) ) ) )
=> ! [V3p: $i] :
( ( mem @ V3p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V3p ) ) ) ) ) ).
thf(ax_thm_2Epair_2EPROD__ALL__def,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b ) @ V0P ) @ V1Q ) @ V2p ) )
<=> ( ( p @ ( ap @ V0P @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2p ) ) )
& ( p @ ( ap @ V1Q @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V2p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPROD__ALL__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b ) @ V0P ) @ V1Q ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2x ) @ V3y ) ) )
<=> ( ( p @ ( ap @ V0P @ V2x ) )
& ( p @ ( ap @ V1Q @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPROD__ALL__MONO,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27b @ bool ) )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ ( arr @ A_27b @ bool ) )
=> ! [V4p: $i] :
( ( mem @ V4p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V5x ) )
=> ( p @ ( ap @ V1P_27 @ V5x ) ) ) )
& ! [V6y: $i] :
( ( mem @ V6y @ A_27b )
=> ( ( p @ ( ap @ V2Q @ V6y ) )
=> ( p @ ( ap @ V3Q_27 @ V6y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b ) @ V0P ) @ V2Q ) @ V4p ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b ) @ V1P_27 ) @ V3Q_27 ) @ V4p ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPROD__ALL__CONG,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1p_27: $i] :
( ( mem @ V1p_27 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ! [V3P_27: $i] :
( ( mem @ V3P_27 @ ( arr @ A_27a @ bool ) )
=> ! [V4Q: $i] :
( ( mem @ V4Q @ ( arr @ A_27b @ bool ) )
=> ! [V5Q_27: $i] :
( ( mem @ V5Q_27 @ ( arr @ A_27b @ bool ) )
=> ( ( ( V0p = V1p_27 )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27b )
=> ( ( V1p_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V6x ) @ V7y ) )
=> ( ( p @ ( ap @ V2P @ V6x ) )
<=> ( p @ ( ap @ V3P_27 @ V6x ) ) ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27b )
=> ( ( V1p_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V8x ) @ V9y ) )
=> ( ( p @ ( ap @ V4Q @ V9y ) )
<=> ( p @ ( ap @ V5Q_27 @ V9y ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b ) @ V2P ) @ V4Q ) @ V0p ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2EPROD__ALL @ A_27a @ A_27b ) @ V3P_27 ) @ V5Q_27 ) @ V1p_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EELIM__PEXISTS,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ? [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1p ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V1p ) ) ) )
<=> ? [V2p1: $i] :
( ( mem @ V2p1 @ A_27a )
& ? [V3p2: $i] :
( ( mem @ V3p2 @ A_27b )
& ( p @ ( ap @ ( ap @ V0P @ V2p1 ) @ V3p2 ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EELIM__PFORALL,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V1p: $i] :
( ( mem @ V1p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( p @ ( ap @ ( ap @ V0P @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V1p ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V1p ) ) ) )
<=> ! [V2p1: $i] :
( ( mem @ V2p1 @ A_27a )
=> ! [V3p2: $i] :
( ( mem @ V3p2 @ A_27b )
=> ( p @ ( ap @ ( ap @ V0P @ V2p1 ) @ V3p2 ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPFORALL__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( p @ ( ap @ ( ap @ V0P @ V1x ) @ V2y ) ) ) )
<=> ( p
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ ( ap @ V0P @ V3x ) @ V4y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPEXISTS__THM,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ? [V2y: $i] :
( ( mem @ V2y @ A_27b )
& ( p @ ( ap @ ( ap @ V0P @ V1x ) @ V2y ) ) ) )
<=> ( p
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ ( ap @ V0P @ V3x ) @ V4y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EELIM__PEXISTS__EVAL,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0P @ V1x ) ) ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27b ) @ ( ap @ V0P @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EELIM__PFORALL__EVAL,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ V0P @ V1x ) ) ) ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27b ) @ ( ap @ V0P @ V2x ) ) ) ) ) ) ).
thf(ax_thm_2Epair_2EPAIR__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 ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1g ) @ V2p )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27c @ A_27d ) @ ( ap @ V0f @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V2p ) ) ) @ ( ap @ V1g @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V2p ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPAIR__MAP__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27c @ A_27d ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27c )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27c @ A_27b @ A_27d ) @ V0f ) @ V1g ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ V2x ) @ V3y ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27d ) @ ( ap @ V0f @ V2x ) ) @ ( ap @ V1g @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EFST__PAIR__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27c ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ A_27d ) )
=> ( ( ap @ ( c_2Epair_2EFST @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ V1f ) @ V2g ) @ V0p ) )
= ( ap @ V1f @ ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ESND__PAIR__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27c ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27b @ A_27d ) )
=> ( ( ap @ ( c_2Epair_2ESND @ A_27c @ A_27d ) @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27b @ A_27c @ A_27d ) @ V1f ) @ V2g ) @ V0p ) )
= ( ap @ V2g @ ( ap @ ( c_2Epair_2ESND @ A_27a @ A_27b ) @ V0p ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ELET2__RAND,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27c @ A_27d ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V2N: $i] :
( ( mem @ V2N @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ( ( ap @ V0P
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( lam @ A_27b
@ ^ [V4y: $i] : ( ap @ ( ap @ V2N @ V3x ) @ V4y ) ) ) ) )
@ V1M ) )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27d )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27d )
@ ( lam @ A_27a
@ ^ [V5x: $i] :
( lam @ A_27b
@ ^ [V6y: $i] : ( ap @ V0P @ ( ap @ ( ap @ V2N @ V5x ) @ V6y ) ) ) ) ) )
@ V1M ) ) ) ) ) ).
thf(conj_thm_2Epair_2ELET2__RATOR,axiom,
! [A_27a1: del,A_27a2: del,A_27b: del,A_27c: del,V0M: $i] :
( ( mem @ V0M @ ( ty_2Epair_2Eprod @ A_27a1 @ A_27a2 ) )
=> ! [V1N: $i] :
( ( mem @ V1N @ ( arr @ A_27a1 @ ( arr @ A_27a2 @ ( arr @ A_27b @ A_27c ) ) ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27b )
=> ( ( ap
@ ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ A_27a1 @ A_27a2 ) @ ( arr @ A_27b @ A_27c ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a1 @ A_27a2 @ ( arr @ A_27b @ A_27c ) )
@ ( lam @ A_27a1
@ ^ [V3x: $i] :
( lam @ A_27a2
@ ^ [V4y: $i] : ( ap @ ( ap @ V1N @ V3x ) @ V4y ) ) ) ) )
@ V0M )
@ V2b )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( ty_2Epair_2Eprod @ A_27a1 @ A_27a2 ) @ A_27c )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a1 @ A_27a2 @ A_27c )
@ ( lam @ A_27a1
@ ^ [V5x: $i] :
( lam @ A_27a2
@ ^ [V6y: $i] : ( ap @ ( ap @ ( ap @ V1N @ V5x ) @ V6y ) @ V2b ) ) ) ) )
@ V0M ) ) ) ) ) ).
thf(conj_thm_2Epair_2Eo__UNCURRY__R,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27d @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ( arr @ A_27b @ A_27d ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27c @ A_27d ) @ V0f ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27d ) @ V1g ) )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ A_27b @ A_27c ) @ ( arr @ A_27b @ A_27d ) ) @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ A_27c @ A_27d ) @ V0f ) ) @ V1g ) ) ) ) ) ).
thf(conj_thm_2Epair_2EC__UNCURRY__L,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ ( arr @ A_27d @ A_27c ) ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27d )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2EC @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27d @ A_27c ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( arr @ A_27d @ A_27c ) ) @ V0f ) ) @ V1x )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27d @ ( arr @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ A_27d @ ( arr @ A_27b @ A_27c ) ) @ ( arr @ A_27b @ ( arr @ A_27d @ A_27c ) ) ) @ ( c_2Ecombin_2EC @ A_27b @ A_27d @ A_27c ) ) @ V0f ) ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Epair_2ES__UNCURRY__R,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ ( arr @ A_27d @ A_27c ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ ( arr @ A_27b @ A_27d ) ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2ES @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ A_27d @ A_27c ) @ V0f ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27d ) @ V1g ) )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2ES @ A_27a @ ( arr @ A_27b @ A_27d ) @ ( arr @ A_27b @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ ( arr @ A_27b @ A_27d ) @ ( arr @ A_27b @ A_27c ) ) @ ( arr @ A_27b @ ( arr @ A_27d @ A_27c ) ) ) @ ( c_2Ecombin_2ES @ A_27b @ A_27d @ A_27c ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ ( arr @ A_27b @ ( arr @ A_27d @ A_27c ) ) @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( arr @ A_27d @ A_27c ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ V0f ) ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) ) ) ) @ V1g ) ) ) ) ) ).
thf(conj_thm_2Epair_2EFORALL__UNCURRY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool ) @ V0f ) ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ bool @ ( arr @ A_27b @ bool ) ) @ ( c_2Ebool_2E_21 @ A_27b ) ) @ V0f ) ) ) ) ) ).
thf(conj_thm_2Epair_2EPAIR__FUN__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) @ bool ) )
=> ( ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) ) )
@ ( lam @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
@ ^ [V1f: $i] : ( ap @ V0P @ V1f ) ) ) )
<=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27c ) ) )
@ ( lam @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27c ) )
@ ^ [V2p: $i] :
( ap @ V0P
@ ( lam @ A_27a
@ ^ [V3a: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Epair_2EFST @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27c ) ) @ V2p ) @ V3a ) ) @ ( ap @ ( ap @ ( c_2Epair_2ESND @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27c ) ) @ V2p ) @ V3a ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Epair_2Epair__CASE__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c ) @ V0p ) @ V1f )
= ( ap @ ( ap @ V1f @ ( ap @ ( c_2Epair_2EFST @ A_27b @ A_27c ) @ V0p ) ) @ ( ap @ ( c_2Epair_2ESND @ A_27b @ A_27c ) @ V0p ) ) ) ) ) ).
thf(conj_thm_2Epair_2Epair__case__thm,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27c )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ V0x ) @ V1y ) ) @ V2f )
= ( ap @ ( ap @ V2f @ V0x ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Epair_2Epair__case__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0x: $i] :
( ( mem @ V0x @ A_27b )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27c )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ V0x ) @ V1y ) ) @ V2f )
= ( ap @ ( ap @ V2f @ V0x ) @ V1y ) ) ) ) ) ).
thf(conj_thm_2Epair_2Epair__case__cong,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f_27: $i] :
( ( mem @ V0f_27 @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
=> ! [V2M_27: $i] :
( ( mem @ V2M_27 @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ( ( ( V1M = V2M_27 )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27c )
=> ( ( V2M_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ V4x ) @ V5y ) )
=> ( ( ap @ ( ap @ V3f @ V4x ) @ V5y )
= ( ap @ ( ap @ V0f_27 @ V4x ) @ V5y ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c ) @ V1M ) @ V3f )
= ( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c ) @ V2M_27 ) @ V0f_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2Epair__case__eq,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ A_27b @ A_27c ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Epair_2Epair__CASE @ A_27a @ A_27b @ A_27c ) @ V0p ) @ V1f )
= V2v )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27b )
& ? [V4y: $i] :
( ( mem @ V4y @ A_27c )
& ( V0p
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27c ) @ V3x ) @ V4y ) )
& ( ( ap @ ( ap @ V1f @ V3x ) @ V4y )
= V2v ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2Edatatype__pair,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0pair: $i] :
( ( mem @ V0pair @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) ) @ A_27c ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ A_27c ) @ ( ap @ V0pair @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) ) ) ) ) ).
thf(ax_thm_2Epair_2ELEX__DEF,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V2s: $i] :
( lam @ A_27b
@ ^ [V3t: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V4u: $i] :
( lam @ A_27b
@ ^ [V5v: $i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ V0R1 @ V2s ) @ V4u ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2s ) @ V4u ) ) @ ( ap @ ( ap @ V1R2 @ V3t ) @ V5v ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ELEX__DEF__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 ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27b )
=> ! [V4c: $i] :
( ( mem @ V4c @ A_27a )
=> ! [V5d: $i] :
( ( mem @ V5d @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V2a ) @ V3b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V4c ) @ V5d ) ) )
<=> ( ( p @ ( ap @ ( ap @ V0R1 @ V2a ) @ V4c ) )
| ( ( V2a = V4c )
& ( p @ ( ap @ ( ap @ V1R2 @ V3b ) @ V5d ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ELEX__MONO,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2R3: $i] :
( ( mem @ V2R3 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V3R4: $i] :
( ( mem @ V3R4 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V5y: $i] :
( ( mem @ V5y @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V6x ) @ V7y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ V6x ) @ V7y ) ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27b )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V2R3 @ V8x ) @ V9y ) )
=> ( p @ ( ap @ ( ap @ V3R4 @ V8x ) @ V9y ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R1 ) @ V2R3 ) @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V1R2 ) @ V3R4 ) @ V4x ) @ V5y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EWF__LEX,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27b ) @ V1Q ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R ) @ V1Q ) ) ) ) ) ) ).
thf(ax_thm_2Epair_2ERPROD__DEF,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Epair_2ERPROD @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 )
= ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V2s: $i] :
( lam @ A_27b
@ ^ [V3t: $i] :
( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ bool )
@ ( lam @ A_27a
@ ^ [V4u: $i] :
( lam @ A_27b
@ ^ [V5v: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V0R1 @ V2s ) @ V4u ) ) @ ( ap @ ( ap @ V1R2 @ V3t ) @ V5v ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EWF__RPROD,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27b ) @ V1Q ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ERPROD @ A_27a @ A_27b ) @ V0R ) @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2Etotal__LEX,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_2Erelation_2Etotal @ A_27a ) @ V0R1 ) )
& ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27b ) @ V1R2 ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etotal @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2Etransitive__LEX,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_2Erelation_2Etransitive @ A_27a ) @ V0R1 ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27b ) @ V1R2 ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2Ereflexive__LEX,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_2Erelation_2Ereflexive @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R1 ) )
| ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27b ) @ V1R2 ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2Esymmetric__LEX,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_2Erelation_2Esymmetric @ A_27a ) @ V0R1 ) )
& ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27b ) @ V1R2 ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ELEX__CONG,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 ) ) )
=> ! [V2v1: $i] :
( ( mem @ V2v1 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V3v2: $i] :
( ( mem @ V3v2 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V4R1_27: $i] :
( ( mem @ V4R1_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V5R2_27: $i] :
( ( mem @ V5R2_27 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V6v1_27: $i] :
( ( mem @ V6v1_27 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V7v2_27: $i] :
( ( mem @ V7v2_27 @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ( ( ( V2v1 = V6v1_27 )
& ( V3v2 = V7v2_27 )
& ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9b: $i] :
( ( mem @ V9b @ A_27b )
=> ! [V10c: $i] :
( ( mem @ V10c @ A_27a )
=> ! [V11d: $i] :
( ( mem @ V11d @ A_27b )
=> ( ( ( V6v1_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V8a ) @ V9b ) )
& ( V7v2_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V10c ) @ V11d ) ) )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V8a ) @ V10c ) )
<=> ( p @ ( ap @ ( ap @ V4R1_27 @ V8a ) @ V10c ) ) ) ) ) ) ) )
& ! [V12a: $i] :
( ( mem @ V12a @ A_27a )
=> ! [V13b: $i] :
( ( mem @ V13b @ A_27b )
=> ! [V14c: $i] :
( ( mem @ V14c @ A_27a )
=> ! [V15d: $i] :
( ( mem @ V15d @ A_27b )
=> ( ( ( V6v1_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V12a ) @ V13b ) )
& ( V7v2_27
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V14c ) @ V15d ) )
& ( V12a = V14c ) )
=> ( ( p @ ( ap @ ( ap @ V1R2 @ V13b ) @ V15d ) )
<=> ( p @ ( ap @ ( ap @ V5R2_27 @ V13b ) @ V15d ) ) ) ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V2v1 ) @ V3v2 ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Epair_2ELEX @ A_27a @ A_27b ) @ V4R1_27 ) @ V5R2_27 ) @ V6v1_27 ) @ V7v2_27 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2EFST__EQ__EQUIV,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ( ap @ ( c_2Epair_2EFST @ A_27a @ A_27b ) @ V0p )
= V1x )
<=> ? [V2y: $i] :
( ( mem @ V2y @ A_27b )
& ( V0p
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27b ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Epair_2ESND__EQ__EQUIV,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( ( ap @ ( c_2Epair_2ESND @ A_27b @ A_27a ) @ V0p )
= V1y )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( V0p
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27a ) @ V2x ) @ V1y ) ) ) ) ) ) ).
%------------------------------------------------------------------------------