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 : 242 ( 108 unt; 78 typ; 0 def)
% Number of atoms : 288 ( 187 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 127 ( 3 ~; 6 |; 32 &)
% ( 40 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 66 ( 40 >; 26 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 76 ( 76 usr; 6 con; 0-7 aty)
% Number of variables : 937 ( 786 !; 24 ?; 937 :)
% ( 127 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2E_23_23_2E0,type,
c_2Epair_2E_23_23_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27c,A_27d)))) ).
tff(c_2Epair_2E_23_23_2E3,type,
c_2Epair_2E_23_23_2E3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27c) * tyop_2Emin_2Efun(A_27b,A_27d) * tyop_2Epair_2Eprod(A_27a,A_27b) ) > tyop_2Epair_2Eprod(A_27c,A_27d) ) ).
tff(c_2Epair_2E_2C_2E0,type,
c_2Epair_2E_2C_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).
tff(c_2Epair_2E_2C_2E2,type,
c_2Epair_2E_2C_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_3F_21_2E0,type,
c_2Ebool_2E_3F_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_21_2E1,type,
c_2Ebool_2E_3F_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2EABS__prod_2E0,type,
c_2Epair_2EABS__prod_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(A_27a,A_27b)) ).
tff(c_2Epair_2EABS__prod_2E1,type,
c_2Epair_2EABS__prod_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Ecombin_2EC_2E0,type,
c_2Ecombin_2EC_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ecombin_2EC_2E2,type,
c_2Ecombin_2EC_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) * A_27b ) > tyop_2Emin_2Efun(A_27a,A_27c) ) ).
tff(c_2Epair_2ECURRY_2E0,type,
c_2Epair_2ECURRY_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))) ).
tff(c_2Epair_2ECURRY_2E1,type,
c_2Epair_2ECURRY_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) ) ).
tff(c_2Epair_2ECURRY_2E3,type,
c_2Epair_2ECURRY_2E3:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c) * A_27a * A_27b ) > A_27c ) ).
tff(c_2Ebool_2EDATATYPE_2E0,type,
c_2Ebool_2EDATATYPE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2EDATATYPE_2E1,type,
c_2Ebool_2EDATATYPE_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Epair_2EFST_2E0,type,
c_2Epair_2EFST_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a) ).
tff(c_2Epair_2EFST_2E1,type,
c_2Epair_2EFST_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27a ) ).
tff(c_2Ebool_2ELET_2E0,type,
c_2Ebool_2ELET_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27b)) ).
tff(c_2Ebool_2ELET_2E2,type,
c_2Ebool_2ELET_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(c_2Epair_2ELEX_2E0,type,
c_2Epair_2ELEX_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))) ).
tff(c_2Epair_2ELEX_2E2,type,
c_2Epair_2ELEX_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)) ) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)) ) ).
tff(c_2Epair_2ELEX_2E4,type,
c_2Epair_2ELEX_2E4:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)) * tyop_2Epair_2Eprod(A_27a,A_27b) * tyop_2Epair_2Eprod(A_27a,A_27b) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2EPROD__ALL_2E0,type,
c_2Epair_2EPROD__ALL_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))) ).
tff(c_2Epair_2EPROD__ALL_2E3,type,
c_2Epair_2EPROD__ALL_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool) * tyop_2Epair_2Eprod(A_27a,A_27b) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2EREP__prod_2E0,type,
c_2Epair_2EREP__prod_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))) ).
tff(c_2Epair_2EREP__prod_2E1,type,
c_2Epair_2EREP__prod_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)) ) ).
tff(c_2Epair_2ERPROD_2E0,type,
c_2Epair_2ERPROD_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))) ).
tff(c_2Epair_2ERPROD_2E2,type,
c_2Epair_2ERPROD_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)) ) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)) ) ).
tff(c_2Ecombin_2ES_2E0,type,
c_2Ecombin_2ES_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ecombin_2ES_2E2,type,
c_2Ecombin_2ES_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) * tyop_2Emin_2Efun(A_27a,A_27b) ) > tyop_2Emin_2Efun(A_27a,A_27c) ) ).
tff(c_2Epair_2ESND_2E0,type,
c_2Epair_2ESND_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b) ).
tff(c_2Epair_2ESND_2E1,type,
c_2Epair_2ESND_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27a,A_27b) > A_27b ) ).
tff(c_2Epair_2ESWAP_2E0,type,
c_2Epair_2ESWAP_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27a,A_27b)) ).
tff(c_2Epair_2ESWAP_2E1,type,
c_2Epair_2ESWAP_2E1:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Epair_2Eprod(A_27b,A_27a) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Epair_2EUNCURRY_2E0,type,
c_2Epair_2EUNCURRY_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)) ).
tff(c_2Epair_2EUNCURRY_2E1,type,
c_2Epair_2EUNCURRY_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c) ) ).
tff(c_2Epair_2EUNCURRY_2E2,type,
c_2Epair_2EUNCURRY_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)) * tyop_2Epair_2Eprod(A_27a,A_27b) ) > A_27c ) ).
tff(c_2Erelation_2EWF_2E0,type,
c_2Erelation_2EWF_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Erelation_2EWF_2E1,type,
c_2Erelation_2EWF_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ecombin_2Eo_2E0,type,
c_2Ecombin_2Eo_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b))) ).
tff(c_2Ecombin_2Eo_2E1,type,
c_2Ecombin_2Eo_2E1:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( tyop_2Emin_2Efun(A_27c,A_27b) > tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(A_27a,A_27b)) ) ).
tff(c_2Ecombin_2Eo_2E2,type,
c_2Ecombin_2Eo_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Emin_2Efun(A_27c,A_27b) * tyop_2Emin_2Efun(A_27a,A_27c) ) > tyop_2Emin_2Efun(A_27a,A_27b) ) ).
tff(c_2Epair_2Epair__CASE_2E0,type,
c_2Epair_2Epair__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)),A_27a)) ).
tff(c_2Epair_2Epair__CASE_2E2,type,
c_2Epair_2Epair__CASE_2E2:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Epair_2Eprod(A_27b,A_27c) * tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)) ) > A_27a ) ).
tff(c_2Erelation_2Ereflexive_2E0,type,
c_2Erelation_2Ereflexive_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Erelation_2Ereflexive_2E1,type,
c_2Erelation_2Ereflexive_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Ebool ) ).
tff(c_2Erelation_2Esymmetric_2E0,type,
c_2Erelation_2Esymmetric_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Erelation_2Esymmetric_2E1,type,
c_2Erelation_2Esymmetric_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Ebool ) ).
tff(c_2Erelation_2Etotal_2E0,type,
c_2Erelation_2Etotal_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Erelation_2Etotal_2E1,type,
c_2Erelation_2Etotal_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Ebool ) ).
tff(c_2Erelation_2Etransitive_2E0,type,
c_2Erelation_2Etransitive_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool) ).
tff(c_2Erelation_2Etransitive_2E1,type,
c_2Erelation_2Etransitive_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq3_2Ec_2Epair_2E_23_23_2E3_2Emono_2EA_27a_20mono_2EA_27c_20mono_2EA_27b_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X1_2E0: tyop_2Emin_2Efun(A_27c,A_27d),X2_2E0: tyop_2Epair_2Eprod(A_27a,A_27c)] : ( c_2Epair_2E_23_23_2E3(A_27a,A_27c,A_27b,A_27d,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d),app_2E2(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27c),tyop_2Epair_2Eprod(A_27b,A_27d))),c_2Epair_2E_23_23_2E0(A_27a,A_27c,A_27b,A_27d),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Epair_2E_23_23_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,A_27c),X1_2E0: tyop_2Emin_2Efun(A_27b,A_27d),X2_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2E_23_23_2E3(A_27a,A_27b,A_27c,A_27d,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27c,A_27d),app_2E2(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27c,A_27d)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Epair_2Eprod(A_27c,A_27d))),c_2Epair_2E_23_23_2E0(A_27a,A_27b,A_27c,A_27d),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27a,X1_2E0: A_27b] : ( c_2Epair_2E_2C_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Epair_2E_2C_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27c: $tType,X0_2E0: A_27a,X1_2E0: A_27c] : ( c_2Epair_2E_2C_2E2(A_27a,A_27c,X0_2E0,X1_2E0) = app_2E2(A_27c,tyop_2Epair_2Eprod(A_27a,A_27c),app_2E2(A_27a,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27a,A_27c)),c_2Epair_2E_2C_2E0(A_27a,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: A_27b,X1_2E0: A_27a] : ( c_2Epair_2E_2C_2E2(A_27b,A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Epair_2Eprod(A_27b,A_27a),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27a)),c_2Epair_2E_2C_2E0(A_27b,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27b: $tType,A_27c: $tType,X0_2E0: A_27b,X1_2E0: A_27c] : ( c_2Epair_2E_2C_2E2(A_27b,A_27c,X0_2E0,X1_2E0) = app_2E2(A_27c,tyop_2Epair_2Eprod(A_27b,A_27c),app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,tyop_2Epair_2Eprod(A_27b,A_27c)),c_2Epair_2E_2C_2E0(A_27b,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27b_20mono_2EA_27d,axiom,
! [A_27b: $tType,A_27d: $tType,X0_2E0: A_27b,X1_2E0: A_27d] : ( c_2Epair_2E_2C_2E2(A_27b,A_27d,X0_2E0,X1_2E0) = app_2E2(A_27d,tyop_2Epair_2Eprod(A_27b,A_27d),app_2E2(A_27b,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27b,A_27d)),c_2Epair_2E_2C_2E0(A_27b,A_27d),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: A_27c,X1_2E0: A_27d] : ( c_2Epair_2E_2C_2E2(A_27c,A_27d,X0_2E0,X1_2E0) = app_2E2(A_27d,tyop_2Epair_2Eprod(A_27c,A_27d),app_2E2(A_27c,tyop_2Emin_2Efun(A_27d,tyop_2Epair_2Eprod(A_27c,A_27d)),c_2Epair_2E_2C_2E0(A_27c,A_27d),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Epair_2Eprod_28A_27b_2CA_27c_29_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c))),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_21_2E1_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Emin_2Efun_28A_27a_2CA_27b_29_2Ctyop_2Emin_2Efun_28A_27a_2CA_27c_29_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_21_2E0(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EABS__prod_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epair_2EABS__prod_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EABS__prod_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2EC_2E2_2Emono_2EA_27a_20mono_2EA_27d_20mono_2Etyop_2Emin_2Efun_28A_27b_2CA_27c_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c))),X1_2E0: A_27d] : ( c_2Ecombin_2EC_2E2(A_27a,A_27d,tyop_2Emin_2Efun(A_27b,A_27c),X0_2E0,X1_2E0) = app_2E2(A_27d,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))),c_2Ecombin_2EC_2E0(A_27a,A_27d,tyop_2Emin_2Efun(A_27b,A_27c)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2EC_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27d_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27d,A_27c)),X1_2E0: A_27d] : ( c_2Ecombin_2EC_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,A_27c,X0_2E0,X1_2E0) = app_2E2(A_27d,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)),c_2Ecombin_2EC_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ECURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)] : ( c_2Epair_2ECURRY_2E1(A_27a,A_27b,A_27c,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),c_2Epair_2ECURRY_2E0(A_27a,A_27b,A_27c),X0_2E0) ) ).
tff(arityeq3_2Ec_2Epair_2ECURRY_2E3_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X1_2E0: A_27a,X2_2E0: A_27b] : ( c_2Epair_2ECURRY_2E3(A_27a,A_27b,A_27c,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),c_2Epair_2ECURRY_2E0(A_27a,A_27b,A_27c),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2EDATATYPE_2E1_2Emono_2EA_27c,axiom,
! [A_27c: $tType,X0_2E0: A_27c] : ( c_2Ebool_2EDATATYPE_2E1(A_27c,X0_2E0) = app_2E2(A_27c,tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E0(A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EFST_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27a,c_2Epair_2EFST_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27b,A_27a)] : ( c_2Epair_2EFST_2E1(A_27b,A_27a,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27b,A_27a),A_27b,c_2Epair_2EFST_2E0(A_27b,A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27b,A_27c)] : ( c_2Epair_2EFST_2E1(A_27b,A_27c,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27b,A_27c),A_27b,c_2Epair_2EFST_2E0(A_27b,A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27c,A_27d)] : ( c_2Epair_2EFST_2E1(A_27c,A_27d,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27c,A_27d),A_27c,c_2Epair_2EFST_2E0(A_27c,A_27d),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EFST_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27a_2CA_27b_29_20mono_2Etyop_2Emin_2Efun_28A_27a_2CA_27c_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))] : ( c_2Epair_2EFST_2E1(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Efun(A_27a,A_27b),c_2Epair_2EFST_2E0(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ELET_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X1_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Ebool_2ELET_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),c_2Ebool_2ELET_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ELET_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),X1_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Ebool_2ELET_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),c_2Ebool_2ELET_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ELET_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a1_2CA_27a2_29_20mono_2EA_27c,axiom,
! [A_27a1: $tType,A_27a2: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a1,A_27a2),A_27c),X1_2E0: tyop_2Epair_2Eprod(A_27a1,A_27a2)] : ( c_2Ebool_2ELET_2E2(tyop_2Epair_2Eprod(A_27a1,A_27a2),A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a1,A_27a2),A_27c,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a1,A_27a2),A_27c),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a1,A_27a2),A_27c),c_2Ebool_2ELET_2E0(tyop_2Epair_2Eprod(A_27a1,A_27a2),A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ELET_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a1_2CA_27a2_29_20mono_2Etyop_2Emin_2Efun_28A_27b_2CA_27c_29,axiom,
! [A_27a1: $tType,A_27a2: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a1,A_27a2),tyop_2Emin_2Efun(A_27b,A_27c)),X1_2E0: tyop_2Epair_2Eprod(A_27a1,A_27a2)] : ( c_2Ebool_2ELET_2E2(tyop_2Epair_2Eprod(A_27a1,A_27a2),tyop_2Emin_2Efun(A_27b,A_27c),X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a1,A_27a2),tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a1,A_27a2),tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a1,A_27a2),tyop_2Emin_2Efun(A_27b,A_27c)),c_2Ebool_2ELET_2E0(tyop_2Epair_2Eprod(A_27a1,A_27a2),tyop_2Emin_2Efun(A_27b,A_27c)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2ELEX_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epair_2ELEX_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epair_2ELEX_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq4_2Ec_2Epair_2ELEX_2E4_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X2_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),X3_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2ELEX_2E4(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epair_2ELEX_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq3_2Ec_2Epair_2EPROD__ALL_2E3_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X2_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EPROD__ALL_2E3(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2EPROD__ALL_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EREP__prod_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EREP__prod_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),c_2Epair_2EREP__prod_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2ERPROD_2E2_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epair_2ERPROD_2E2(A_27a,A_27b,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),c_2Epair_2ERPROD_2E0(A_27a,A_27b),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2ES_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Efun_28A_27b_2CA_27d_29_20mono_2Etyop_2Emin_2Efun_28A_27b_2CA_27c_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c))),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d))] : ( c_2Ecombin_2ES_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))),c_2Ecombin_2ES_2E0(A_27a,tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2ES_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27d_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27d,A_27c)),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d)] : ( c_2Ecombin_2ES_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)),c_2Ecombin_2ES_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2ESND_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27b,c_2Epair_2ESND_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2EA_27b_20mono_2EA_27a,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27b,A_27a)] : ( c_2Epair_2ESND_2E1(A_27b,A_27a,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27b,A_27a),A_27a,c_2Epair_2ESND_2E0(A_27b,A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27b,A_27c)] : ( c_2Epair_2ESND_2E1(A_27b,A_27c,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27b,A_27c),A_27c,c_2Epair_2ESND_2E0(A_27b,A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27c,A_27d)] : ( c_2Epair_2ESND_2E1(A_27c,A_27d,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27c,A_27d),A_27d,c_2Epair_2ESND_2E0(A_27c,A_27d),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESND_2E1_2Emono_2Etyop_2Emin_2Efun_28A_27a_2CA_27b_29_20mono_2Etyop_2Emin_2Efun_28A_27a_2CA_27c_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))] : ( c_2Epair_2ESND_2E1(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),X0_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Efun(A_27a,A_27c),c_2Epair_2ESND_2E0(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2ESWAP_2E1_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27b,A_27a)] : ( c_2Epair_2ESWAP_2E1(A_27a,A_27b,X0_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27b,A_27a),tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2ESWAP_2E0(A_27a,A_27b),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29_29_20mono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,A_27c),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2EUNCURRY_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),X1_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27d,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,A_27d),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Emin_2Ebool,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Ebool,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,tyop_2Emin_2Ebool),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Emin_2Efun_28A_27d_2CA_27c_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Efun(A_27d,A_27c),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27d,A_27c)),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a_20mono_2EA_27b_20mono_2Etyop_2Emin_2Efun_28tyop_2Epair_2Eprod_28A_27a_2CA_27b_29_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),c_2Epair_2EUNCURRY_2E0(A_27a,A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a1_20mono_2EA_27a2_20mono_2EA_27c,axiom,
! [A_27a1: $tType,A_27a2: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,A_27c))] : ( c_2Epair_2EUNCURRY_2E1(A_27a1,A_27a2,A_27c,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,A_27c)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a1,A_27a2),A_27c),c_2Epair_2EUNCURRY_2E0(A_27a1,A_27a2,A_27c),X0_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2EA_27a1_20mono_2EA_27a2_20mono_2Etyop_2Emin_2Efun_28A_27b_2CA_27c_29,axiom,
! [A_27a1: $tType,A_27a2: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c)))] : ( c_2Epair_2EUNCURRY_2E1(A_27a1,A_27a2,tyop_2Emin_2Efun(A_27b,A_27c),X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a1,A_27a2),tyop_2Emin_2Efun(A_27b,A_27c)),c_2Epair_2EUNCURRY_2E0(A_27a1,A_27a2,tyop_2Emin_2Efun(A_27b,A_27c)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2EWF_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] : ( c_2Erelation_2EWF_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2EWF_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ecombin_2Eo_2E1_2Emono_2EA_27b_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27d,A_27c)] : ( c_2Ecombin_2Eo_2E1(A_27b,A_27c,A_27d,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27d,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c)),c_2Ecombin_2Eo_2E0(A_27b,A_27c,A_27d),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29_20mono_2EA_27c_20mono_2EA_27d,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(A_27d,A_27c),X1_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d)] : ( c_2Ecombin_2Eo_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,A_27d,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),app_2E2(tyop_2Emin_2Efun(A_27d,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)),c_2Ecombin_2Eo_2E0(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,A_27d),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Efun_28A_27b_2CA_27c_29_20mono_2Etyop_2Emin_2Efun_28A_27b_2CA_27d_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c)),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d))] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(A_27b,A_27d),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(A_27b,A_27d)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Ebool_20mono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Ebool_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Efun_28A_27d_2Ctyop_2Emin_2Efun_28A_27b_2CA_27c_29_29_20mono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Efun_28A_27d_2CA_27c_29_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c))),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)))] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c))),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c)))),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Efun_28tyop_2Emin_2Efun_28A_27b_2CA_27d_29_2Ctyop_2Emin_2Efun_28A_27b_2CA_27c_29_29_20mono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Efun_28A_27d_2CA_27c_29_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c))),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)))] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c))),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c)))),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecombin_2Eo_2E2_2Emono_2EA_27a_20mono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Emin_2Efun_28A_27d_2CA_27c_29_29_20mono_2Etyop_2Emin_2Efun_28A_27b_2Ctyop_2Epair_2Eprod_28A_27a_2CA_27b_29_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)))] : ( c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)))),c_2Ecombin_2Eo_2E0(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ecombin_2Eo_2E1_2Emono_2EA_27b_20mono_2Etyop_2Emin_2Efun_28A_27d_2CA_27c_29_20mono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27d,A_27c))] : ( c_2Ecombin_2Eo_2E1(A_27b,tyop_2Emin_2Efun(A_27d,A_27c),tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c))),c_2Ecombin_2Eo_2E0(A_27b,tyop_2Emin_2Efun(A_27d,A_27c),tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2Epair__CASE_2E2_2Emono_2EA_27a_20mono_2EA_27b_20mono_2EA_27c,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,X0_2E0: tyop_2Epair_2Eprod(A_27b,A_27c),X1_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a))] : ( c_2Epair_2Epair__CASE_2E2(A_27a,A_27b,A_27c,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)),A_27a,app_2E2(tyop_2Epair_2Eprod(A_27b,A_27c),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)),A_27a),c_2Epair_2Epair__CASE_2E0(A_27a,A_27b,A_27c),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Ereflexive_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Ereflexive_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Ereflexive_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Ereflexive_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Ereflexive_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Ereflexive_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Ereflexive_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] : ( c_2Erelation_2Ereflexive_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Ereflexive_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Esymmetric_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Esymmetric_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Esymmetric_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Esymmetric_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Esymmetric_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Esymmetric_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Esymmetric_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] : ( c_2Erelation_2Esymmetric_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Esymmetric_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Etotal_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Etotal_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Etotal_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Etotal_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Etotal_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Etotal_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Etotal_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] : ( c_2Erelation_2Etotal_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Etotal_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Etransitive_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Etransitive_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Etransitive_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Etransitive_2E1_2Emono_2EA_27b,axiom,
! [A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Erelation_2Etransitive_2E1(A_27b,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Etransitive_2E0(A_27b),X0_2E0) ) ).
tff(arityeq1_2Ec_2Erelation_2Etransitive_2E1_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2CA_27b_29,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))] : ( c_2Erelation_2Etransitive_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,c_2Erelation_2Etransitive_2E0(tyop_2Epair_2Eprod(A_27a,A_27b)),X0_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Epair_2Eprod__TY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)] :
( ! [V1p_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool,F0_2E0,V1p_2E0))
<=> ? [V2x_2E0: A_27a,V3y_2E0: A_27b] :
! [V4x_27_2E0: A_27a,V5x_27_27_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1p_2E0,V4x_27_2E0),V5x_27_27_2E0))
<=> ( ( V4x_27_2E0 = V2x_2E0 )
& ( V5x_27_27_2E0 = V3y_2E0 ) ) ) )
=> ? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Epair_2Eprod(A_27a,A_27b),F0_2E0,V0rep_2E0)) ) ).
tff(thm_2Epair_2EABS__REP__prod,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EABS__prod_2E1(A_27a,A_27b,c_2Epair_2EREP__prod_2E1(A_27a,A_27b,V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ? [V2x_2E0: A_27a,V3y_2E0: A_27b] :
! [V4x_27_2E0: A_27a,V5x_27_27_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1r_2E0,V4x_27_2E0),V5x_27_27_2E0))
<=> ( ( V4x_27_2E0 = V2x_2E0 )
& ( V5x_27_27_2E0 = V3y_2E0 ) ) )
<=> ( c_2Epair_2EREP__prod_2E1(A_27a,A_27b,c_2Epair_2EABS__prod_2E1(A_27a,A_27b,V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Epair_2ECOMMA__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))] :
( ! [V0x_2E0: A_27a,V1y_2E0: A_27b,V2a_2E0: A_27a,V3b_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1y_2E0),V2a_2E0),V3b_2E0))
<=> ( ( V2a_2E0 = V0x_2E0 )
& ( V3b_2E0 = V1y_2E0 ) ) )
=> ! [V0x_2E0: A_27a,V1y_2E0: A_27b] : ( c_2Epair_2E_2C_2E2(A_27a,A_27b,V0x_2E0,V1y_2E0) = c_2Epair_2EABS__prod_2E1(A_27a,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),F0_2E0,V0x_2E0),V1y_2E0)) ) ) ).
tff(thm_2Epair_2EPAIR,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2E_2C_2E2(A_27a,A_27b,c_2Epair_2EFST_2E1(A_27a,A_27b,V0x_2E0),c_2Epair_2ESND_2E1(A_27a,A_27b,V0x_2E0)) = V0x_2E0 ) ).
tff(thm_2Epair_2ESWAP__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0a_2E0: tyop_2Epair_2Eprod(A_27b,A_27a)] : ( c_2Epair_2ESWAP_2E1(A_27a,A_27b,V0a_2E0) = c_2Epair_2E_2C_2E2(A_27a,A_27b,c_2Epair_2ESND_2E1(A_27b,A_27a,V0a_2E0),c_2Epair_2EFST_2E1(A_27b,A_27a,V0a_2E0)) ) ).
tff(thm_2Epair_2ECURRY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),V1x_2E0: A_27a,V2y_2E0: A_27b] : ( c_2Epair_2ECURRY_2E3(A_27a,A_27b,A_27c,V0f_2E0,V1x_2E0,V2y_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,V0f_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V1x_2E0,V2y_2E0)) ) ).
tff(thm_2Epair_2EUNCURRY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1v_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,V0f_2E0,V1v_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V1v_2E0)),c_2Epair_2ESND_2E1(A_27a,A_27b,V1v_2E0)) ) ).
tff(thm_2Epair_2EPROD__ALL__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
( p(c_2Epair_2EPROD__ALL_2E3(A_27a,A_27b,V0P_2E0,V1Q_2E0,V2p_2E0))
<=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V2p_2E0)))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1Q_2E0,c_2Epair_2ESND_2E1(A_27a,A_27b,V2p_2E0))) ) ) ).
tff(thm_2Epair_2EPAIR__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V1g_2E0: tyop_2Emin_2Efun(A_27b,A_27d),V2p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2E_23_23_2E3(A_27a,A_27b,A_27c,A_27d,V0f_2E0,V1g_2E0,V2p_2E0) = c_2Epair_2E_2C_2E2(A_27c,A_27d,app_2E2(A_27a,A_27c,V0f_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V2p_2E0)),app_2E2(A_27b,A_27d,V1g_2E0,c_2Epair_2ESND_2E1(A_27a,A_27b,V2p_2E0))) ) ).
tff(thm_2Epair_2Epair__CASE__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0p_2E0: tyop_2Epair_2Eprod(A_27b,A_27c),V1f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a))] : ( c_2Epair_2Epair__CASE_2E2(A_27a,A_27b,A_27c,V0p_2E0,V1f_2E0) = app_2E2(A_27c,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,A_27a),V1f_2E0,c_2Epair_2EFST_2E1(A_27b,A_27c,V0p_2E0)),c_2Epair_2ESND_2E1(A_27b,A_27c,V0p_2E0)) ) ).
tff(thm_2Epair_2ELEX__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))))] :
( ! [V0R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2s_2E0: A_27a,V1R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3t_2E0: A_27b,V4u_2E0: A_27a,V5v_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))),F1_2E0,V0R1_2E0),V2s_2E0),V1R2_2E0),V3t_2E0),V4u_2E0),V5v_2E0))
<=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R1_2E0,V2s_2E0),V4u_2E0))
| ( ( V2s_2E0 = V4u_2E0 )
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R2_2E0,V3t_2E0),V5v_2E0)) ) ) )
=> ( ! [V0R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2s_2E0: A_27a,V3t_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F0_2E0,V0R1_2E0),V1R2_2E0),V2s_2E0),V3t_2E0) = c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))),F1_2E0,V0R1_2E0),V2s_2E0),V1R2_2E0),V3t_2E0)) )
=> ! [V0R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epair_2ELEX_2E2(A_27a,A_27b,V0R1_2E0,V1R2_2E0) = c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F0_2E0,V0R1_2E0),V1R2_2E0)) ) ) ) ).
tff(thm_2Epair_2ERPROD__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))))] :
( ! [V0R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V2s_2E0: A_27a,V1R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3t_2E0: A_27b,V4u_2E0: A_27a,V5v_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))),F1_2E0,V0R1_2E0),V2s_2E0),V1R2_2E0),V3t_2E0),V4u_2E0),V5v_2E0))
<=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R1_2E0,V2s_2E0),V4u_2E0))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R2_2E0,V3t_2E0),V5v_2E0)) ) )
=> ( ! [V0R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2s_2E0: A_27a,V3t_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F0_2E0,V0R1_2E0),V1R2_2E0),V2s_2E0),V3t_2E0) = c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))))),F1_2E0,V0R1_2E0),V2s_2E0),V1R2_2E0),V3t_2E0)) )
=> ! [V0R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Epair_2ERPROD_2E2(A_27a,A_27b,V0R1_2E0,V1R2_2E0) = c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)))),F0_2E0,V0R1_2E0),V1R2_2E0)) ) ) ) ).
tff(thm_2Epair_2EPAIR__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0y_2E0: A_27b,V1x_2E0: A_27a,V2b_2E0: A_27b,V3a_2E0: A_27a] :
( ( c_2Epair_2E_2C_2E2(A_27a,A_27b,V1x_2E0,V0y_2E0) = c_2Epair_2E_2C_2E2(A_27a,A_27b,V3a_2E0,V2b_2E0) )
<=> ( ( V1x_2E0 = V3a_2E0 )
& ( V0y_2E0 = V2b_2E0 ) ) ) ).
tff(thm_2Epair_2ECLOSED__PAIR__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b,V2a_2E0: A_27a,V3b_2E0: A_27b] :
( ( c_2Epair_2E_2C_2E2(A_27a,A_27b,V0x_2E0,V1y_2E0) = c_2Epair_2E_2C_2E2(A_27a,A_27b,V2a_2E0,V3b_2E0) )
<=> ( ( V0x_2E0 = V2a_2E0 )
& ( V1y_2E0 = V3b_2E0 ) ) ) ).
tff(thm_2Epair_2EABS__PAIR__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
? [V1q_2E0: A_27a,V2r_2E0: A_27b] : ( V0x_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V1q_2E0,V2r_2E0) ) ).
tff(thm_2Epair_2Epair__CASES,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
? [V1q_2E0: A_27a,V2r_2E0: A_27b] : ( V0x_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V1q_2E0,V2r_2E0) ) ).
tff(thm_2Epair_2EFST,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( c_2Epair_2EFST_2E1(A_27a,A_27b,c_2Epair_2E_2C_2E2(A_27a,A_27b,V0x_2E0,V1y_2E0)) = V0x_2E0 ) ).
tff(thm_2Epair_2ESND,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( c_2Epair_2ESND_2E1(A_27a,A_27b,c_2Epair_2E_2C_2E2(A_27a,A_27b,V0x_2E0,V1y_2E0)) = V1y_2E0 ) ).
tff(thm_2Epair_2EPAIR__FST__SND__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V1q_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
( ( V0p_2E0 = V1q_2E0 )
<=> ( ( c_2Epair_2EFST_2E1(A_27a,A_27b,V0p_2E0) = c_2Epair_2EFST_2E1(A_27a,A_27b,V1q_2E0) )
& ( c_2Epair_2ESND_2E1(A_27a,A_27b,V0p_2E0) = c_2Epair_2ESND_2E1(A_27a,A_27b,V1q_2E0) ) ) ) ).
tff(thm_2Epair_2EUNCURRY__VAR,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1v_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,V0f_2E0,V1v_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V1v_2E0)),c_2Epair_2ESND_2E1(A_27a,A_27b,V1v_2E0)) ) ).
tff(thm_2Epair_2EELIM__UNCURRY,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,V0f_2E0,V1x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V1x_2E0)),c_2Epair_2ESND_2E1(A_27a,A_27b,V1x_2E0)) ) ).
tff(thm_2Epair_2EUNCURRY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1x_2E0: A_27a,V2y_2E0: A_27b] : ( c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,V0f_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V1x_2E0,V2y_2E0)) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V1x_2E0),V2y_2E0) ) ).
tff(thm_2Epair_2ECURRY__UNCURRY__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))] : ( c_2Epair_2ECURRY_2E1(A_27a,A_27b,A_27c,c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,V0f_2E0)) = V0f_2E0 ) ).
tff(thm_2Epair_2EUNCURRY__CURRY__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)] : ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,c_2Epair_2ECURRY_2E1(A_27a,A_27b,A_27c,V0f_2E0)) = V0f_2E0 ) ).
tff(thm_2Epair_2ECURRY__ONE__ONE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),V1f_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)] :
( ( c_2Epair_2ECURRY_2E1(A_27a,A_27b,A_27c,V1f_2E0) = c_2Epair_2ECURRY_2E1(A_27a,A_27b,A_27c,V0g_2E0) )
<=> ( V1f_2E0 = V0g_2E0 ) ) ).
tff(thm_2Epair_2EUNCURRY__ONE__ONE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))] :
( ( c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,V1f_2E0) = c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,V0g_2E0) )
<=> ( V1f_2E0 = V0g_2E0 ) ) ).
tff(thm_2Epair_2Epair__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))] :
? [V1fn_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c)] :
! [V2x_2E0: A_27a,V3y_2E0: A_27b] : ( app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,V1fn_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V2x_2E0,V3y_2E0)) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),V3y_2E0) ) ).
tff(thm_2Epair_2EUNCURRY__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V2M_27_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V3M_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
( ( ( V3M_2E0 = V2M_27_2E0 )
& ! [V4x_2E0: A_27a,V5y_2E0: A_27b] :
( ( V2M_27_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V4x_2E0,V5y_2E0) )
=> ( app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V1f_2E0,V4x_2E0),V5y_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_27_2E0,V4x_2E0),V5y_2E0) ) ) )
=> ( c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,V1f_2E0,V3M_2E0) = c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,V0f_27_2E0,V2M_27_2E0) ) ) ).
tff(thm_2Epair_2ELAMBDA__PROD,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),V2p1_2E0: A_27a,V3p2_2E0: A_27b] : ( app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),F0_2E0,V0P_2E0),V2p1_2E0),V3p2_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,V0P_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V2p1_2E0,V3p2_2E0)) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),V1x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : ( app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,V0P_2E0,V1x_2E0) = c_2Epair_2EUNCURRY_2E2(A_27a,A_27b,A_27c,app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),F0_2E0,V0P_2E0),V1x_2E0) ) ) ).
tff(thm_2Epair_2EEXISTS__PROD,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)] :
( ? [V1p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,V1p_2E0))
<=> ? [V2p__1_2E0: A_27a,V3p__2_2E0: A_27b] : p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V2p__1_2E0,V3p__2_2E0))) ) ).
tff(thm_2Epair_2EFORALL__PROD,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)] :
( ! [V1p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,V1p_2E0))
<=> ! [V2p__1_2E0: A_27a,V3p__2_2E0: A_27b] : p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V2p__1_2E0,V3p__2_2E0))) ) ).
tff(thm_2Epair_2Epair__induction,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool)] :
( ! [V1p__1_2E0: A_27a,V2p__2_2E0: A_27b] : p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V1p__1_2E0,V2p__2_2E0)))
=> ! [V3p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : p(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Ebool,V0P_2E0,V3p_2E0)) ) ).
tff(thm_2Epair_2EPROD__ALL__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0y_2E0: A_27b,V1x_2E0: A_27a,V2Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Epair_2EPROD__ALL_2E3(A_27a,A_27b,V3P_2E0,V2Q_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V1x_2E0,V0y_2E0)))
<=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V3P_2E0,V1x_2E0))
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,V2Q_2E0,V0y_2E0)) ) ) ).
tff(thm_2Epair_2EPROD__ALL__MONO,axiom,
! [A_27a: $tType,A_27b: $tType,V0p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V1Q_27_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3P_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( ( ! [V5x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V4P_2E0,V5x_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V3P_27_2E0,V5x_2E0)) )
& ! [V6y_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,V2Q_2E0,V6y_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,V1Q_27_2E0,V6y_2E0)) ) )
=> ( p(c_2Epair_2EPROD__ALL_2E3(A_27a,A_27b,V4P_2E0,V2Q_2E0,V0p_2E0))
=> p(c_2Epair_2EPROD__ALL_2E3(A_27a,A_27b,V3P_27_2E0,V1Q_27_2E0,V0p_2E0)) ) ) ).
tff(thm_2Epair_2EPROD__ALL__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V1p_27_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V2P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V3P_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V5Q_27_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)] :
( ( ( V0p_2E0 = V1p_27_2E0 )
& ! [V6x_2E0: A_27a,V7y_2E0: A_27b] :
( ( V1p_27_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V6x_2E0,V7y_2E0) )
=> ( app_2E2(A_27a,tyop_2Emin_2Ebool,V2P_2E0,V6x_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,V3P_27_2E0,V6x_2E0) ) )
& ! [V8x_2E0: A_27a,V9y_2E0: A_27b] :
( ( V1p_27_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V8x_2E0,V9y_2E0) )
=> ( app_2E2(A_27b,tyop_2Emin_2Ebool,V4Q_2E0,V9y_2E0) = app_2E2(A_27b,tyop_2Emin_2Ebool,V5Q_27_2E0,V9y_2E0) ) ) )
=> ( c_2Epair_2EPROD__ALL_2E3(A_27a,A_27b,V2P_2E0,V4Q_2E0,V0p_2E0) = c_2Epair_2EPROD__ALL_2E3(A_27a,A_27b,V3P_27_2E0,V5Q_27_2E0,V1p_27_2E0) ) ) ).
tff(thm_2Epair_2EELIM__PEXISTS,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ? [V1p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V1p_2E0)),c_2Epair_2ESND_2E1(A_27a,A_27b,V1p_2E0)))
<=> ? [V2p1_2E0: A_27a,V3p2_2E0: A_27b] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V2p1_2E0),V3p2_2E0)) ) ).
tff(thm_2Epair_2EELIM__PFORALL,axiom,
! [A_27a: $tType,A_27b: $tType,V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ! [V1p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V1p_2E0)),c_2Epair_2ESND_2E1(A_27a,A_27b,V1p_2E0)))
<=> ! [V2p1_2E0: A_27a,V3p2_2E0: A_27b] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V2p1_2E0),V3p2_2E0)) ) ).
tff(thm_2Epair_2EPFORALL__THM,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3x_2E0: A_27a,V4y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0),V3x_2E0),V4y_2E0) = app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V3x_2E0),V4y_2E0) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ! [V1x_2E0: A_27a,V2y_2E0: A_27b] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V1x_2E0),V2y_2E0))
<=> p(c_2Ebool_2E_21_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0)))) ) ) ).
tff(thm_2Epair_2EPEXISTS__THM,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3x_2E0: A_27a,V4y_2E0: A_27b] : ( app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0),V3x_2E0),V4y_2E0) = app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V3x_2E0),V4y_2E0) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ? [V1x_2E0: A_27a,V2y_2E0: A_27b] : p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V1x_2E0),V2y_2E0))
<=> p(c_2Ebool_2E_3F_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0)))) ) ) ).
tff(thm_2Epair_2EELIM__PEXISTS__EVAL,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0),V1x_2E0) = app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V1x_2E0) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( p(c_2Ebool_2E_3F_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0))))
<=> ? [V2x_2E0: A_27a] : p(c_2Ebool_2E_3F_2E1(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V2x_2E0))) ) ) ).
tff(thm_2Epair_2EELIM__PFORALL__EVAL,axiom,
! [A_27a: $tType,A_27b: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1x_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0),V1x_2E0) = app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V1x_2E0) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( p(c_2Ebool_2E_21_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),F0_2E0,V0P_2E0))))
<=> ! [V2x_2E0: A_27a] : p(c_2Ebool_2E_21_2E1(A_27b,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V0P_2E0,V2x_2E0))) ) ) ).
tff(thm_2Epair_2EPAIR__MAP__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27c,A_27d),V2x_2E0: A_27a,V3y_2E0: A_27c] : ( c_2Epair_2E_23_23_2E3(A_27a,A_27c,A_27b,A_27d,V0f_2E0,V1g_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27c,V2x_2E0,V3y_2E0)) = c_2Epair_2E_2C_2E2(A_27b,A_27d,app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0),app_2E2(A_27c,A_27d,V1g_2E0,V3y_2E0)) ) ).
tff(thm_2Epair_2EFST__PAIR__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V2g_2E0: tyop_2Emin_2Efun(A_27b,A_27d)] : ( c_2Epair_2EFST_2E1(A_27c,A_27d,c_2Epair_2E_23_23_2E3(A_27a,A_27b,A_27c,A_27d,V1f_2E0,V2g_2E0,V0p_2E0)) = app_2E2(A_27a,A_27c,V1f_2E0,c_2Epair_2EFST_2E1(A_27a,A_27b,V0p_2E0)) ) ).
tff(thm_2Epair_2ESND__PAIR__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V1f_2E0: tyop_2Emin_2Efun(A_27a,A_27c),V2g_2E0: tyop_2Emin_2Efun(A_27b,A_27d)] : ( c_2Epair_2ESND_2E1(A_27c,A_27d,c_2Epair_2E_23_23_2E3(A_27a,A_27b,A_27c,A_27d,V1f_2E0,V2g_2E0,V0p_2E0)) = app_2E2(A_27b,A_27d,V2g_2E0,c_2Epair_2ESND_2E1(A_27a,A_27b,V0p_2E0)) ) ).
tff(thm_2Epair_2ELET2__RAND,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)))] :
( ! [V0P_2E0: tyop_2Emin_2Efun(A_27c,A_27d),V2N_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V5x_2E0: A_27a,V6y_2E0: A_27b] : ( app_2E2(A_27b,A_27d,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27d),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),app_2E2(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d))),F1_2E0,V0P_2E0),V2N_2E0),V5x_2E0),V6y_2E0) = app_2E2(A_27c,A_27d,V0P_2E0,app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V2N_2E0,V5x_2E0),V6y_2E0)) )
=> ( ! [V2N_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V3x_2E0: A_27a,V4y_2E0: A_27b] : ( app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),F0_2E0,V2N_2E0),V3x_2E0),V4y_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V2N_2E0,V3x_2E0),V4y_2E0) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(A_27c,A_27d),V1M_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V2N_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c))] : ( app_2E2(A_27c,A_27d,V0P_2E0,c_2Ebool_2ELET_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),F0_2E0,V2N_2E0)),V1M_2E0)) = c_2Ebool_2ELET_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27d,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),app_2E2(tyop_2Emin_2Efun(A_27c,A_27d),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d))),F1_2E0,V0P_2E0),V2N_2E0)),V1M_2E0) ) ) ) ).
tff(thm_2Epair_2ELET2__RATOR,axiom,
! [A_27a1: $tType,A_27a2: $tType,A_27b: $tType,A_27c: $tType,F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,A_27c)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))))] :
( ! [V1N_2E0: tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),V2b_2E0: A_27b,V5x_2E0: A_27a1,V6y_2E0: A_27a2] : ( app_2E2(A_27a2,A_27c,app_2E2(A_27a1,tyop_2Emin_2Efun(A_27a2,A_27c),app_2E2(A_27b,tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,A_27c)),app_2E2(tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,A_27c))),F1_2E0,V1N_2E0),V2b_2E0),V5x_2E0),V6y_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c)),V1N_2E0,V5x_2E0),V6y_2E0),V2b_2E0) )
=> ( ! [V1N_2E0: tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),V3x_2E0: A_27a1,V4y_2E0: A_27a2] : ( app_2E2(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c)),app_2E2(tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),F0_2E0,V1N_2E0),V3x_2E0),V4y_2E0) = app_2E2(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c)),V1N_2E0,V3x_2E0),V4y_2E0) )
=> ! [V0M_2E0: tyop_2Epair_2Eprod(A_27a1,A_27a2),V1N_2E0: tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),V2b_2E0: A_27b] : ( app_2E2(A_27b,A_27c,c_2Ebool_2ELET_2E2(tyop_2Epair_2Eprod(A_27a1,A_27a2),tyop_2Emin_2Efun(A_27b,A_27c),c_2Epair_2EUNCURRY_2E1(A_27a1,A_27a2,tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),F0_2E0,V1N_2E0)),V0M_2E0),V2b_2E0) = c_2Ebool_2ELET_2E2(tyop_2Epair_2Eprod(A_27a1,A_27a2),A_27c,c_2Epair_2EUNCURRY_2E1(A_27a1,A_27a2,A_27c,app_2E2(A_27b,tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,A_27c)),app_2E2(tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,tyop_2Emin_2Efun(A_27b,A_27c))),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27a1,tyop_2Emin_2Efun(A_27a2,A_27c))),F1_2E0,V1N_2E0),V2b_2E0)),V0M_2E0) ) ) ) ).
tff(thm_2Epair_2Eo__UNCURRY__R,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),V1f_2E0: tyop_2Emin_2Efun(A_27d,A_27c)] : ( c_2Ecombin_2Eo_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27c,A_27d,V1f_2E0,c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27d,V0g_2E0)) = c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),tyop_2Emin_2Efun(A_27b,A_27d),c_2Ecombin_2Eo_2E1(A_27b,A_27c,A_27d,V1f_2E0),V0g_2E0)) ) ).
tff(thm_2Epair_2EC__UNCURRY__L,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0x_2E0: A_27d,V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)))] : ( c_2Ecombin_2EC_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,A_27c,c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Efun(A_27d,A_27c),V1f_2E0),V0x_2E0) = c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,c_2Ecombin_2EC_2E2(A_27a,A_27d,tyop_2Emin_2Efun(A_27b,A_27c),c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Efun(A_27d,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),c_2Ecombin_2EC_2E0(A_27b,A_27d,A_27c),V1f_2E0),V0x_2E0)) ) ).
tff(thm_2Epair_2ES__UNCURRY__R,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27d)),V1f_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27b),tyop_2Emin_2Efun(A_27d,A_27c))] : ( c_2Ecombin_2ES_2E2(tyop_2Epair_2Eprod(A_27a,A_27b),A_27d,A_27c,V1f_2E0,c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27d,V0g_2E0)) = c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,A_27c,c_2Ecombin_2ES_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c),c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27d),tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),c_2Ecombin_2ES_2E0(A_27b,A_27d,A_27c),c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27d,A_27c)),tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b)),c_2Ecombin_2Eo_2E1(A_27b,tyop_2Emin_2Efun(A_27d,A_27c),tyop_2Epair_2Eprod(A_27a,A_27b),V1f_2E0),c_2Epair_2E_2C_2E0(A_27a,A_27b))),V0g_2E0)) ) ).
tff(thm_2Epair_2EFORALL__UNCURRY,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] : ( c_2Ebool_2E_21_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2EUNCURRY_2E1(A_27a,A_27b,tyop_2Emin_2Ebool,V0f_2E0)) = c_2Ebool_2E_21_2E1(A_27a,c_2Ecombin_2Eo_2E2(A_27a,tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),c_2Ebool_2E_21_2E0(A_27b),V0f_2E0)) ) ).
tff(thm_2Epair_2EPAIR__FUN__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,F2_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c))),F1_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool))] :
( ! [V2p_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),V3a_2E0: A_27a] : ( app_2E2(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c),app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),F2_2E0,V2p_2E0),V3a_2E0) = c_2Epair_2E_2C_2E2(A_27b,A_27c,app_2E2(A_27a,A_27b,c_2Epair_2EFST_2E1(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),V2p_2E0),V3a_2E0),app_2E2(A_27a,A_27c,c_2Epair_2ESND_2E1(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),V2p_2E0),V3a_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),V2p_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))] : ( app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Ebool),F1_2E0,V0P_2E0),V2p_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool,V0P_2E0,app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),F2_2E0,V2p_2E0)) )
=> ( ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c))] : ( app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),F0_2E0,V0P_2E0),V1f_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool,V0P_2E0,V1f_2E0) )
=> ! [V0P_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_21_2E1(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),F0_2E0,V0P_2E0)) = c_2Ebool_2E_3F_21_2E1(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27b,A_27c)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),tyop_2Emin_2Ebool),F1_2E0,V0P_2E0)) ) ) ) ) ).
tff(thm_2Epair_2Epair__case__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0y_2E0: A_27c,V1x_2E0: A_27b,V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a))] : ( c_2Epair_2Epair__CASE_2E2(A_27a,A_27b,A_27c,c_2Epair_2E_2C_2E2(A_27b,A_27c,V1x_2E0,V0y_2E0),V2f_2E0) = app_2E2(A_27c,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,A_27a),V2f_2E0,V1x_2E0),V0y_2E0) ) ).
tff(thm_2Epair_2Epair__case__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0y_2E0: A_27c,V1x_2E0: A_27b,V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a))] : ( c_2Epair_2Epair__CASE_2E2(A_27a,A_27b,A_27c,c_2Epair_2E_2C_2E2(A_27b,A_27c,V1x_2E0,V0y_2E0),V2f_2E0) = app_2E2(A_27c,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,A_27a),V2f_2E0,V1x_2E0),V0y_2E0) ) ).
tff(thm_2Epair_2Epair__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_27_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a)),V1M_2E0: tyop_2Epair_2Eprod(A_27b,A_27c),V2M_27_2E0: tyop_2Epair_2Eprod(A_27b,A_27c),V3f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a))] :
( ( ( V1M_2E0 = V2M_27_2E0 )
& ! [V4x_2E0: A_27b,V5y_2E0: A_27c] :
( ( V2M_27_2E0 = c_2Epair_2E_2C_2E2(A_27b,A_27c,V4x_2E0,V5y_2E0) )
=> ( app_2E2(A_27c,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,A_27a),V3f_2E0,V4x_2E0),V5y_2E0) = app_2E2(A_27c,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,A_27a),V0f_27_2E0,V4x_2E0),V5y_2E0) ) ) )
=> ( c_2Epair_2Epair__CASE_2E2(A_27a,A_27b,A_27c,V1M_2E0,V3f_2E0) = c_2Epair_2Epair__CASE_2E2(A_27a,A_27b,A_27c,V2M_27_2E0,V0f_27_2E0) ) ) ).
tff(thm_2Epair_2Epair__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v_2E0: A_27a,V1p_2E0: tyop_2Epair_2Eprod(A_27b,A_27c),V2f_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27c,A_27a))] :
( ( c_2Epair_2Epair__CASE_2E2(A_27a,A_27b,A_27c,V1p_2E0,V2f_2E0) = V0v_2E0 )
<=> ? [V3x_2E0: A_27b,V4y_2E0: A_27c] :
( ( V1p_2E0 = c_2Epair_2E_2C_2E2(A_27b,A_27c,V3x_2E0,V4y_2E0) )
& ( app_2E2(A_27c,A_27a,app_2E2(A_27b,tyop_2Emin_2Efun(A_27c,A_27a),V2f_2E0,V3x_2E0),V4y_2E0) = V0v_2E0 ) ) ) ).
tff(thm_2Epair_2Edatatype__pair,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0pair_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))),A_27c)] : p(c_2Ebool_2EDATATYPE_2E1(A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))),A_27c,V0pair_2E0,c_2Epair_2E_2C_2E0(A_27a,A_27b)))) ).
tff(thm_2Epair_2ELEX__DEF__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0d_2E0: A_27b,V1c_2E0: A_27a,V2b_2E0: A_27b,V3a_2E0: A_27a,V4R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V5R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( p(c_2Epair_2ELEX_2E4(A_27a,A_27b,V5R1_2E0,V4R2_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27b,V3a_2E0,V2b_2E0),c_2Epair_2E_2C_2E2(A_27a,A_27b,V1c_2E0,V0d_2E0)))
<=> ( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5R1_2E0,V3a_2E0),V1c_2E0))
| ( ( V3a_2E0 = V1c_2E0 )
& p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V4R2_2E0,V2b_2E0),V0d_2E0)) ) ) ) ).
tff(thm_2Epair_2ELEX__MONO,axiom,
! [A_27a: $tType,A_27b: $tType,V0y_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V1x_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V2R4_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V3R3_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V4R2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V5R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ( ! [V6x_2E0: A_27a,V7y_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V5R1_2E0,V6x_2E0),V7y_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4R2_2E0,V6x_2E0),V7y_2E0)) )
& ! [V8x_2E0: A_27b,V9y_2E0: A_27b] :
( p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V3R3_2E0,V8x_2E0),V9y_2E0))
=> p(app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V2R4_2E0,V8x_2E0),V9y_2E0)) ) )
=> ( p(c_2Epair_2ELEX_2E4(A_27a,A_27b,V5R1_2E0,V3R3_2E0,V1x_2E0,V0y_2E0))
=> p(c_2Epair_2ELEX_2E4(A_27a,A_27b,V4R2_2E0,V2R4_2E0,V1x_2E0,V0y_2E0)) ) ) ).
tff(thm_2Epair_2EWF__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ( p(c_2Erelation_2EWF_2E1(A_27a,V0R_2E0))
& p(c_2Erelation_2EWF_2E1(A_27b,V1Q_2E0)) )
=> p(c_2Erelation_2EWF_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2ELEX_2E2(A_27a,A_27b,V0R_2E0,V1Q_2E0))) ) ).
tff(thm_2Epair_2EWF__RPROD,axiom,
! [A_27a: $tType,A_27b: $tType,V0R_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1Q_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool))] :
( ( p(c_2Erelation_2EWF_2E1(A_27a,V0R_2E0))
& p(c_2Erelation_2EWF_2E1(A_27b,V1Q_2E0)) )
=> p(c_2Erelation_2EWF_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2ERPROD_2E2(A_27a,A_27b,V0R_2E0,V1Q_2E0))) ) ).
tff(thm_2Epair_2Etotal__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ( p(c_2Erelation_2Etotal_2E1(A_27a,V1R1_2E0))
& p(c_2Erelation_2Etotal_2E1(A_27b,V0R2_2E0)) )
=> p(c_2Erelation_2Etotal_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2ELEX_2E2(A_27a,A_27b,V1R1_2E0,V0R2_2E0))) ) ).
tff(thm_2Epair_2Etransitive__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ( p(c_2Erelation_2Etransitive_2E1(A_27a,V1R1_2E0))
& p(c_2Erelation_2Etransitive_2E1(A_27b,V0R2_2E0)) )
=> p(c_2Erelation_2Etransitive_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2ELEX_2E2(A_27a,A_27b,V1R1_2E0,V0R2_2E0))) ) ).
tff(thm_2Epair_2Ereflexive__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( p(c_2Erelation_2Ereflexive_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2ELEX_2E2(A_27a,A_27b,V1R1_2E0,V0R2_2E0)))
<=> ( p(c_2Erelation_2Ereflexive_2E1(A_27a,V1R1_2E0))
| p(c_2Erelation_2Ereflexive_2E1(A_27b,V0R2_2E0)) ) ) ).
tff(thm_2Epair_2Esymmetric__LEX,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V1R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))] :
( ( p(c_2Erelation_2Esymmetric_2E1(A_27a,V1R1_2E0))
& p(c_2Erelation_2Esymmetric_2E1(A_27b,V0R2_2E0)) )
=> p(c_2Erelation_2Esymmetric_2E1(tyop_2Epair_2Eprod(A_27a,A_27b),c_2Epair_2ELEX_2E2(A_27a,A_27b,V1R1_2E0,V0R2_2E0))) ) ).
tff(thm_2Epair_2ELEX__CONG,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V1R2_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V2v1_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V3v2_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V4R1_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),V5R2_27_2E0: tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool)),V6v1_27_2E0: tyop_2Epair_2Eprod(A_27a,A_27b),V7v2_27_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
( ( ( V2v1_2E0 = V6v1_27_2E0 )
& ( V3v2_2E0 = V7v2_27_2E0 )
& ! [V8a_2E0: A_27a,V9b_2E0: A_27b,V10c_2E0: A_27a,V11d_2E0: A_27b] :
( ( ( V6v1_27_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V8a_2E0,V9b_2E0) )
& ( V7v2_27_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V10c_2E0,V11d_2E0) ) )
=> ( app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0R1_2E0,V8a_2E0),V10c_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V4R1_27_2E0,V8a_2E0),V10c_2E0) ) )
& ! [V12a_2E0: A_27a,V13b_2E0: A_27b,V14c_2E0: A_27a,V15d_2E0: A_27b] :
( ( ( V6v1_27_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V12a_2E0,V13b_2E0) )
& ( V7v2_27_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V14c_2E0,V15d_2E0) )
& ( V12a_2E0 = V14c_2E0 ) )
=> ( app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V1R2_2E0,V13b_2E0),V15d_2E0) = app_2E2(A_27b,tyop_2Emin_2Ebool,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Ebool),V5R2_27_2E0,V13b_2E0),V15d_2E0) ) ) )
=> ( c_2Epair_2ELEX_2E4(A_27a,A_27b,V0R1_2E0,V1R2_2E0,V2v1_2E0,V3v2_2E0) = c_2Epair_2ELEX_2E4(A_27a,A_27b,V4R1_27_2E0,V5R2_27_2E0,V6v1_27_2E0,V7v2_27_2E0) ) ) ).
tff(thm_2Epair_2EFST__EQ__EQUIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1p_2E0: tyop_2Epair_2Eprod(A_27a,A_27b)] :
( ( c_2Epair_2EFST_2E1(A_27a,A_27b,V1p_2E0) = V0x_2E0 )
<=> ? [V2y_2E0: A_27b] : ( V1p_2E0 = c_2Epair_2E_2C_2E2(A_27a,A_27b,V0x_2E0,V2y_2E0) ) ) ).
tff(thm_2Epair_2ESND__EQ__EQUIV,axiom,
! [A_27a: $tType,A_27b: $tType,V0y_2E0: A_27a,V1p_2E0: tyop_2Epair_2Eprod(A_27b,A_27a)] :
( ( c_2Epair_2ESND_2E1(A_27b,A_27a,V1p_2E0) = V0y_2E0 )
<=> ? [V2x_2E0: A_27b] : ( V1p_2E0 = c_2Epair_2E_2C_2E2(A_27b,A_27a,V2x_2E0,V0y_2E0) ) ) ).
%------------------------------------------------------------------------------