ITP001 Axioms: ITP121_7.ax
%------------------------------------------------------------------------------
% File : ITP121_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 : metric.ax [Gau19]
% : HL4121_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 150 ( 47 unt; 71 typ; 0 def)
% Number of atoms : 136 ( 63 equ)
% Maximal formula atoms : 6 ( 0 avg)
% Number of connectives : 62 ( 5 ~; 3 |; 10 &)
% ( 21 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 55 ( 34 >; 21 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 67 ( 67 usr; 14 con; 0-4 aty)
% Number of variables : 269 ( 212 !; 6 ?; 269 :)
% ( 51 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Emetric_2Emetric,type,
tyop_2Emetric_2Emetric: $tType > $tType ).
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
tff(tyop_2Etopology_2Etopology,type,
tyop_2Etopology_2Etopology: $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_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_2Enum_2E0_2E0,type,
c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).
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_2Emetric_2EB_2E0,type,
c_2Emetric_2EB_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool))) ).
tff(c_2Emetric_2EB_2E2,type,
c_2Emetric_2EB_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emetric_2Emetric(A_27a) * tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal) ) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) ).
tff(c_2Emetric_2EB_2E3,type,
c_2Emetric_2EB_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emetric_2Emetric(A_27a) * tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal) * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
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_2Epred__set_2EUNIV_2E0,type,
c_2Epred__set_2EUNIV_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,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_2Ereal_2Eabs_2E0,type,
c_2Ereal_2Eabs_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Eabs_2E1,type,
c_2Ereal_2Eabs_2E1: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
tff(c_2Emetric_2Edist_2E0,type,
c_2Emetric_2Edist_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal)) ).
tff(c_2Emetric_2Edist_2E1,type,
c_2Emetric_2Edist_2E1:
!>[A_27a: $tType] : ( tyop_2Emetric_2Emetric(A_27a) > tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal) ) ).
tff(c_2Emetric_2Edist_2E2,type,
c_2Emetric_2Edist_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emetric_2Emetric(A_27a) * tyop_2Epair_2Eprod(A_27a,A_27a) ) > tyop_2Erealax_2Ereal ) ).
tff(c_2Emetric_2Eismet_2E0,type,
c_2Emetric_2Eismet_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool) ).
tff(c_2Emetric_2Eismet_2E1,type,
c_2Emetric_2Eismet_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal) > tyop_2Emin_2Ebool ) ).
tff(c_2Etopology_2Eistopology_2E0,type,
c_2Etopology_2Eistopology_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Etopology_2Eistopology_2E1,type,
c_2Etopology_2Eistopology_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Etopology_2Elimpt_2E0,type,
c_2Etopology_2Elimpt_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))) ).
tff(c_2Etopology_2Elimpt_2E3,type,
c_2Etopology_2Elimpt_2E3:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology(A_27a) * A_27a * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emetric_2Emetric_2E0,type,
c_2Emetric_2Emetric_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emetric_2Emetric(A_27a)) ).
tff(c_2Emetric_2Emetric_2E1,type,
c_2Emetric_2Emetric_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal) > tyop_2Emetric_2Emetric(A_27a) ) ).
tff(c_2Emetric_2Emr1_2E0,type,
c_2Emetric_2Emr1_2E0: tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal) ).
tff(c_2Emetric_2Emtop_2E0,type,
c_2Emetric_2Emtop_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Etopology_2Etopology(A_27a)) ).
tff(c_2Emetric_2Emtop_2E1,type,
c_2Emetric_2Emtop_2E1:
!>[A_27a: $tType] : ( tyop_2Emetric_2Emetric(A_27a) > tyop_2Etopology_2Etopology(A_27a) ) ).
tff(c_2Etopology_2Eneigh_2E0,type,
c_2Etopology_2Eneigh_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Etopology_2Eneigh_2E2,type,
c_2Etopology_2Eneigh_2E2:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology(A_27a) * tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Etopology_2Eopen__in_2E0,type,
c_2Etopology_2Eopen__in_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)) ).
tff(c_2Etopology_2Eopen__in_2E2,type,
c_2Etopology_2Eopen__in_2E2:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology(A_27a) * tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Erealax_2Ereal__add_2E0,type,
c_2Erealax_2Ereal__add_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Erealax_2Ereal__add_2E2,type,
c_2Erealax_2Ereal__add_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Erealax_2Ereal__lt_2E0,type,
c_2Erealax_2Ereal__lt_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Erealax_2Ereal__lt_2E2,type,
c_2Erealax_2Ereal__lt_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Ereal__lte_2E0,type,
c_2Ereal_2Ereal__lte_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)) ).
tff(c_2Ereal_2Ereal__lte_2E2,type,
c_2Ereal_2Ereal__lte_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Emin_2Ebool ).
tff(c_2Ereal_2Ereal__of__num_2E0,type,
c_2Ereal_2Ereal__of__num_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal) ).
tff(c_2Ereal_2Ereal__of__num_2E1,type,
c_2Ereal_2Ereal__of__num_2E1: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
tff(c_2Ereal_2Ereal__sub_2E0,type,
c_2Ereal_2Ereal__sub_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)) ).
tff(c_2Ereal_2Ereal__sub_2E2,type,
c_2Ereal_2Ereal__sub_2E2: ( tyop_2Erealax_2Ereal * tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
tff(c_2Etopology_2Etopology_2E0,type,
c_2Etopology_2Etopology_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(A_27a)) ).
tff(c_2Etopology_2Etopology_2E1,type,
c_2Etopology_2Etopology_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) > tyop_2Etopology_2Etopology(A_27a) ) ).
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(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] : ( c_2Epair_2E_2C_2E2(A_27a,A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Epair_2Eprod(A_27a,A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(A_27a,A_27a)),c_2Epair_2E_2C_2E0(A_27a,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C_2E2(A_27a,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal)),c_2Epair_2E_2C_2E0(A_27a,tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Emin_2Efun_28A_27a_2Ctyop_2Emin_2Ebool_29_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X1_2E0: A_27a] : ( c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a)),c_2Epair_2E_2C_2E0(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Epair_2E_2C_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),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(arityeq2_2Ec_2Emetric_2EB_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emetric_2Emetric(A_27a),X1_2E0: tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal)] : ( c_2Emetric_2EB_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Emetric_2EB_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Emetric_2EB_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emetric_2Emetric(A_27a),X1_2E0: tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal),X2_2E0: A_27a] : ( c_2Emetric_2EB_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Emetric_2EB_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Epair_2Eprod_28A_27a_2CA_27a_29_2Ctyop_2Erealax_2Ereal_29_20mono_2Etyop_2Emetric_2Emetric_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emetric_2Emetric(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal)),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emetric_2Emetric(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Epair_2EUNCURRY_2E1_2Emono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal_20mono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] : ( c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Epair_2EUNCURRY_2E0(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),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_2Ereal_2Eabs_2E1,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs_2E1(X0_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,c_2Ereal_2Eabs_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Emetric_2Edist_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emetric_2Emetric(A_27a)] : ( c_2Emetric_2Edist_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),c_2Emetric_2Edist_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Emetric_2Edist_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emetric_2Emetric(A_27a),X1_2E0: tyop_2Epair_2Eprod(A_27a,A_27a)] : ( c_2Emetric_2Edist_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),c_2Emetric_2Edist_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Emetric_2Edist_2E2_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)] : ( c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal,app_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),c_2Emetric_2Edist_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Emetric_2Eismet_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal)] : ( c_2Emetric_2Eismet_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Emetric_2Eismet_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emetric_2Eismet_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)] : ( c_2Emetric_2Eismet_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),tyop_2Emin_2Ebool,c_2Emetric_2Eismet_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Etopology_2Eistopology_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eistopology_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Etopology_2Eistopology_2E0(A_27a),X0_2E0) ) ).
tff(arityeq3_2Ec_2Etopology_2Elimpt_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Etopology_2Etopology(A_27a),X1_2E0: A_27a,X2_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Elimpt_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Etopology_2Elimpt_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Etopology_2Elimpt_2E3_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),X1_2E0: tyop_2Erealax_2Ereal,X2_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Elimpt_2E3(tyop_2Erealax_2Ereal,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),app_2E2(tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)),c_2Etopology_2Elimpt_2E0(tyop_2Erealax_2Ereal),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Emetric_2Emetric_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal)] : ( c_2Emetric_2Emetric_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emetric_2Emetric(A_27a),c_2Emetric_2Emetric_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emetric_2Emetric_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal)] : ( c_2Emetric_2Emetric_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),tyop_2Erealax_2Ereal),tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),c_2Emetric_2Emetric_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emetric_2Emtop_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emetric_2Emetric(A_27a)] : ( c_2Emetric_2Emtop_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Etopology_2Etopology(A_27a),c_2Emetric_2Emtop_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Emetric_2Emtop_2E1_2Emono_2Etyop_2Erealax_2Ereal,axiom,
! [X0_2E0: tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal)] : ( c_2Emetric_2Emtop_2E1(tyop_2Erealax_2Ereal,X0_2E0) = app_2E2(tyop_2Emetric_2Emetric(tyop_2Erealax_2Ereal),tyop_2Etopology_2Etopology(tyop_2Erealax_2Ereal),c_2Emetric_2Emtop_2E0(tyop_2Erealax_2Ereal),X0_2E0) ) ).
tff(arityeq2_2Ec_2Etopology_2Eneigh_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Etopology_2Etopology(A_27a),X1_2E0: tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a)] : ( c_2Etopology_2Eneigh_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a),tyop_2Emin_2Ebool),c_2Etopology_2Eneigh_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Etopology_2Eopen__in_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Etopology_2Etopology(A_27a),X1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Etopology_2Eopen__in_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Etopology_2Etopology(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Etopology_2Eopen__in_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__add_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__add_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Erealax_2Ereal__lt_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Erealax_2Ereal__lt_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__lte_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),c_2Ereal_2Ereal__lte_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ereal_2Ereal__of__num_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Ereal_2Ereal__of__num_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Erealax_2Ereal,c_2Ereal_2Ereal__of__num_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ereal_2Ereal__sub_2E2,axiom,
! [X0_2E0: tyop_2Erealax_2Ereal,X1_2E0: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__sub_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Ereal_2Ereal__sub_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Etopology_2Etopology_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool)] : ( c_2Etopology_2Etopology_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Etopology_2Etopology(A_27a),c_2Etopology_2Etopology_2E0(A_27a),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_2Emetric_2Eismet,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal)] :
( p(c_2Emetric_2Eismet_2E1(A_27a,V0m_2E0))
<=> ( ! [V1x_2E0: A_27a,V2y_2E0: A_27a] :
( ( app_2E2(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V2y_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V1x_2E0 = V2y_2E0 ) )
& ! [V3x_2E0: A_27a,V4y_2E0: A_27a,V5z_2E0: A_27a] : p(c_2Ereal_2Ereal__lte_2E2(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V4y_2E0,V5z_2E0)),c_2Erealax_2Ereal__add_2E2(app_2E2(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V3x_2E0,V4y_2E0)),app_2E2(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V3x_2E0,V5z_2E0))))) ) ) ).
tff(thm_2Emetric_2Emetric__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal),tyop_2Emetric_2Emetric(A_27a),c_2Emetric_2Eismet_2E0(A_27a),V0rep_2E0)) ).
tff(thm_2Emetric_2Emetric__tybij,axiom,
! [A_27a: $tType] :
( ! [V0a_2E0: tyop_2Emetric_2Emetric(A_27a)] : ( c_2Emetric_2Emetric_2E1(A_27a,c_2Emetric_2Edist_2E1(A_27a,V0a_2E0)) = V0a_2E0 )
& ! [V1r_2E0: tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,A_27a),tyop_2Erealax_2Ereal)] :
( p(c_2Emetric_2Eismet_2E1(A_27a,V1r_2E0))
<=> ( c_2Emetric_2Edist_2E1(A_27a,c_2Emetric_2Emetric_2E1(A_27a,V1r_2E0)) = V1r_2E0 ) ) ) ).
tff(thm_2Emetric_2Emtop,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] :
( ! [V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1S_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V0m_2E0),V1S_27_2E0))
<=> ! [V2x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1S_27_2E0,V2x_2E0))
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4y_2E0: A_27a] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V2x_2E0,V4y_2E0)),V3e_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1S_27_2E0,V4y_2E0)) ) ) ) )
=> ! [V0m_2E0: tyop_2Emetric_2Emetric(A_27a)] : ( c_2Emetric_2Emtop_2E1(A_27a,V0m_2E0) = c_2Etopology_2Etopology_2E1(A_27a,app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V0m_2E0)) ) ) ).
tff(thm_2Emetric_2Eball,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2e_2E0: tyop_2Erealax_2Ereal,V3x_27_2E0: A_27a] : ( c_2Emetric_2EB_2E3(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0),V3x_27_2E0) = c_2Erealax_2Ereal__lt_2E2(c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V3x_27_2E0)),V2e_2E0) ) ).
tff(thm_2Emetric_2Emr1,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1y_2E0) = c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0)) )
=> ( c_2Emetric_2Emr1_2E0 = c_2Emetric_2Emetric_2E1(tyop_2Erealax_2Ereal,c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0)) ) ) ).
tff(thm_2Emetric_2EMETRIC__ISMET,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a)] : p(c_2Emetric_2Eismet_2E1(A_27a,c_2Emetric_2Edist_2E1(A_27a,V0m_2E0))) ).
tff(thm_2Emetric_2EMETRIC__ZERO,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( ( c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V2y_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) )
<=> ( V1x_2E0 = V2y_2E0 ) ) ).
tff(thm_2Emetric_2EMETRIC__SAME,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a] : ( c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V1x_2E0)) = c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0) ) ).
tff(thm_2Emetric_2EMETRIC__POS,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2y_2E0: A_27a] : p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V2y_2E0)))) ).
tff(thm_2Emetric_2EMETRIC__SYM,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2y_2E0: A_27a] : ( c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V2y_2E0)) = c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V2y_2E0,V1x_2E0)) ) ).
tff(thm_2Emetric_2EMETRIC__TRIANGLE,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2y_2E0: A_27a,V3z_2E0: A_27a] : p(c_2Ereal_2Ereal__lte_2E2(c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V3z_2E0)),c_2Erealax_2Ereal__add_2E2(c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V2y_2E0)),c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V2y_2E0,V3z_2E0))))) ).
tff(thm_2Emetric_2EMETRIC__NZ,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2y_2E0: A_27a] :
( ( V1x_2E0 != V2y_2E0 )
=> p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V2y_2E0)))) ) ).
tff(thm_2Emetric_2Emtop__istopology,axiom,
! [A_27a: $tType,F0_2E0: tyop_2Emin_2Efun(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool))] :
( ! [V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1S_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V0m_2E0),V1S_27_2E0))
<=> ! [V2x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1S_27_2E0,V2x_2E0))
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4y_2E0: A_27a] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V2x_2E0,V4y_2E0)),V3e_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V1S_27_2E0,V4y_2E0)) ) ) ) )
=> ! [V0m_2E0: tyop_2Emetric_2Emetric(A_27a)] : p(c_2Etopology_2Eistopology_2E1(A_27a,app_2E2(tyop_2Emetric_2Emetric(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0,V0m_2E0))) ) ).
tff(thm_2Emetric_2EMTOP__OPEN,axiom,
! [A_27a: $tType,V0S_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V1m_2E0: tyop_2Emetric_2Emetric(A_27a)] :
( p(c_2Etopology_2Eopen__in_2E2(A_27a,c_2Emetric_2Emtop_2E1(A_27a,V1m_2E0),V0S_27_2E0))
<=> ! [V2x_2E0: A_27a] :
( p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0S_27_2E0,V2x_2E0))
=> ? [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
& ! [V4y_2E0: A_27a] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Emetric_2Edist_2E2(A_27a,V1m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V2x_2E0,V4y_2E0)),V3e_2E0))
=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0S_27_2E0,V4y_2E0)) ) ) ) ) ).
tff(thm_2Emetric_2EBALL__OPEN,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> p(c_2Etopology_2Eopen__in_2E2(A_27a,c_2Emetric_2Emtop_2E1(A_27a,V0m_2E0),c_2Emetric_2EB_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0)))) ) ).
tff(thm_2Emetric_2EBALL__NEIGH,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V2e_2E0))
=> p(c_2Etopology_2Eneigh_2E2(A_27a,c_2Emetric_2Emtop_2E1(A_27a,V0m_2E0),c_2Epair_2E_2C_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),A_27a,c_2Emetric_2EB_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,tyop_2Erealax_2Ereal,V1x_2E0,V2e_2E0)),V1x_2E0))) ) ).
tff(thm_2Emetric_2EMTOP__LIMPT,axiom,
! [A_27a: $tType,V0m_2E0: tyop_2Emetric_2Emetric(A_27a),V1x_2E0: A_27a,V2S_27_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Etopology_2Elimpt_2E3(A_27a,c_2Emetric_2Emtop_2E1(A_27a,V0m_2E0),V1x_2E0,V2S_27_2E0))
<=> ! [V3e_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V3e_2E0))
=> ? [V4y_2E0: A_27a] :
( ( V1x_2E0 != V4y_2E0 )
& p(app_2E2(A_27a,tyop_2Emin_2Ebool,V2S_27_2E0,V4y_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Emetric_2Edist_2E2(A_27a,V0m_2E0,c_2Epair_2E_2C_2E2(A_27a,A_27a,V1x_2E0,V4y_2E0)),V3e_2E0)) ) ) ) ).
tff(thm_2Emetric_2EISMET__R1,axiom,
! [F0_2E0: tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal))] :
( ! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( app_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,app_2E2(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),F0_2E0,V0x_2E0),V1y_2E0) = c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0)) )
=> p(c_2Emetric_2Eismet_2E1(tyop_2Erealax_2Ereal,c_2Epair_2EUNCURRY_2E1(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,F0_2E0))) ) ).
tff(thm_2Emetric_2EMR1__DEF,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal] : ( c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)) = c_2Ereal_2Eabs_2E1(c_2Ereal_2Ereal__sub_2E2(V1y_2E0,V0x_2E0)) ) ).
tff(thm_2Emetric_2EMR1__ADD,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal] : ( c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1d_2E0))) = c_2Ereal_2Eabs_2E1(V1d_2E0) ) ).
tff(thm_2Emetric_2EMR1__SUB,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal] : ( c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1d_2E0))) = c_2Ereal_2Eabs_2E1(V1d_2E0) ) ).
tff(thm_2Emetric_2EMR1__ADD__POS,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1d_2E0))
=> ( c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1d_2E0))) = V1d_2E0 ) ) ).
tff(thm_2Emetric_2EMR1__SUB__LE,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Ereal_2Ereal__lte_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1d_2E0))
=> ( c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1d_2E0))) = V1d_2E0 ) ) ).
tff(thm_2Emetric_2EMR1__ADD__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1d_2E0))
=> ( c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Erealax_2Ereal__add_2E2(V0x_2E0,V1d_2E0))) = V1d_2E0 ) ) ).
tff(thm_2Emetric_2EMR1__SUB__LT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1d_2E0: tyop_2Erealax_2Ereal] :
( p(c_2Erealax_2Ereal__lt_2E2(c_2Ereal_2Ereal__of__num_2E1(c_2Enum_2E0_2E0),V1d_2E0))
=> ( c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,c_2Ereal_2Ereal__sub_2E2(V0x_2E0,V1d_2E0))) = V1d_2E0 ) ) ).
tff(thm_2Emetric_2EMR1__BETWEEN1,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal,V1y_2E0: tyop_2Erealax_2Ereal,V2z_2E0: tyop_2Erealax_2Ereal] :
( ( p(c_2Erealax_2Ereal__lt_2E2(V0x_2E0,V2z_2E0))
& p(c_2Erealax_2Ereal__lt_2E2(c_2Emetric_2Edist_2E2(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0,c_2Epair_2E_2C_2E2(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal,V0x_2E0,V1y_2E0)),c_2Ereal_2Ereal__sub_2E2(V2z_2E0,V0x_2E0))) )
=> p(c_2Erealax_2Ereal__lt_2E2(V1y_2E0,V2z_2E0)) ) ).
tff(thm_2Emetric_2EMR1__LIMPT,axiom,
! [V0x_2E0: tyop_2Erealax_2Ereal] : p(c_2Etopology_2Elimpt_2E3(tyop_2Erealax_2Ereal,c_2Emetric_2Emtop_2E1(tyop_2Erealax_2Ereal,c_2Emetric_2Emr1_2E0),V0x_2E0,c_2Epred__set_2EUNIV_2E0(tyop_2Erealax_2Ereal))) ).
%------------------------------------------------------------------------------