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 : 71 ( 17 unt; 36 typ; 0 def)
% Number of atoms : 81 ( 21 equ; 3 cnn)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 489 ( 3 ~; 1 |; 7 &; 453 @)
% ( 13 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg; 453 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 78 ( 78 >; 0 *; 0 +; 0 <<)
% Number of symbols : 35 ( 33 usr; 3 con; 0-5 aty)
% Number of variables : 134 ( 13 ^ 96 !; 4 ?; 134 :)
% ( 21 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emetric_2Emetric,type,
tyop_2Emetric_2Emetric: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(tyop_2Etopology_2Etopology,type,
tyop_2Etopology_2Etopology: $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emetric_2EB,type,
c_2Emetric_2EB:
!>[A_27a: $tType] : ( ( tyop_2Emetric_2Emetric @ A_27a ) > ( tyop_2Epair_2Eprod @ A_27a @ tyop_2Erealax_2Ereal ) > A_27a > $o ) ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Emetric_2Edist,type,
c_2Emetric_2Edist:
!>[A_27a: $tType] : ( ( tyop_2Emetric_2Emetric @ A_27a ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > tyop_2Erealax_2Ereal ) ).
thf(c_2Emetric_2Eismet,type,
c_2Emetric_2Eismet:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > tyop_2Erealax_2Ereal ) > $o ) ).
thf(c_2Etopology_2Eistopology,type,
c_2Etopology_2Eistopology:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > $o ) ).
thf(c_2Etopology_2Elimpt,type,
c_2Etopology_2Elimpt:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Emetric_2Emetric,type,
c_2Emetric_2Emetric:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > tyop_2Erealax_2Ereal ) > ( tyop_2Emetric_2Emetric @ A_27a ) ) ).
thf(c_2Emetric_2Emr1,type,
c_2Emetric_2Emr1: tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ).
thf(c_2Emetric_2Emtop,type,
c_2Emetric_2Emtop:
!>[A_27a: $tType] : ( ( tyop_2Emetric_2Emetric @ A_27a ) > ( tyop_2Etopology_2Etopology @ A_27a ) ) ).
thf(c_2Etopology_2Eneigh,type,
c_2Etopology_2Eneigh:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ A_27a ) > $o ) ).
thf(c_2Etopology_2Eopen__in,type,
c_2Etopology_2Eopen__in:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > ( A_27a > $o ) > $o ) ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etopology_2Etopology,type,
c_2Etopology_2Etopology:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > ( tyop_2Etopology_2Etopology @ A_27a ) ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Emetric_2Eismet,axiom,
! [A_27a: $tType,V0m: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > tyop_2Erealax_2Ereal] :
( ( c_2Emetric_2Eismet @ A_27a @ V0m )
<=> ( ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V1x = V2y ) )
& ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] : ( c_2Ereal_2Ereal__lte @ ( V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4y @ V5z ) ) @ ( c_2Erealax_2Ereal__add @ ( V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3x @ V4y ) ) @ ( V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3x @ V5z ) ) ) ) ) ) ).
thf(thm_2Emetric_2Emetric__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Emetric_2Emetric @ A_27a ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > tyop_2Erealax_2Ereal] : ( c_2Ebool_2ETYPE__DEFINITION @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > tyop_2Erealax_2Ereal ) @ ( tyop_2Emetric_2Emetric @ A_27a ) @ ( c_2Emetric_2Eismet @ A_27a ) @ V0rep ) ).
thf(thm_2Emetric_2Emetric__tybij,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Emetric_2Emetric @ A_27a] :
( ( c_2Emetric_2Emetric @ A_27a @ ( c_2Emetric_2Edist @ A_27a @ V0a ) )
= V0a )
& ! [V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > tyop_2Erealax_2Ereal] :
( ( c_2Emetric_2Eismet @ A_27a @ V1r )
<=> ( ( c_2Emetric_2Edist @ A_27a @ ( c_2Emetric_2Emetric @ A_27a @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Emetric_2Emtop,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a] :
( ( c_2Emetric_2Emtop @ A_27a @ V0m )
= ( c_2Etopology_2Etopology @ A_27a
@ ^ [V1S_27: A_27a > $o] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2x: A_27a] :
( c_2Emin_2E_3D_3D_3E @ ( V1S_27 @ V2x )
@ ( c_2Ebool_2E_3F @ tyop_2Erealax_2Ereal
@ ^ [V3e: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V4y ) ) @ V3e ) @ ( V1S_27 @ V4y ) ) ) ) ) ) ) ) ) ).
thf(thm_2Emetric_2Eball,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Emetric_2EB @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ tyop_2Erealax_2Ereal @ V1x @ V2e ) )
= ( ^ [V3y: A_27a] : ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V3y ) ) @ V2e ) ) ) ).
thf(thm_2Emetric_2Emr1,axiom,
( c_2Emetric_2Emr1
= ( c_2Emetric_2Emetric @ tyop_2Erealax_2Ereal
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) ) ) ) ).
thf(thm_2Emetric_2EMETRIC__ISMET,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a] : ( c_2Emetric_2Eismet @ A_27a @ ( c_2Emetric_2Edist @ A_27a @ V0m ) ) ).
thf(thm_2Emetric_2EMETRIC__ZERO,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
<=> ( V1x = V2y ) ) ).
thf(thm_2Emetric_2EMETRIC__SAME,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a] :
( ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V1x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Emetric_2EMETRIC__POS,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2y: A_27a] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) ) ) ).
thf(thm_2Emetric_2EMETRIC__SYM,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) )
= ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2y @ V1x ) ) ) ).
thf(thm_2Emetric_2EMETRIC__TRIANGLE,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] : ( c_2Ereal_2Ereal__lte @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V3z ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) ) @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2y @ V3z ) ) ) ) ).
thf(thm_2Emetric_2EMETRIC__NZ,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2y: A_27a] :
( ( (~) @ ( V1x = V2y ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) ) ) ) ).
thf(thm_2Emetric_2Emtop__istopology,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a] :
( c_2Etopology_2Eistopology @ A_27a
@ ^ [V1S_27: A_27a > $o] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V2x: A_27a] :
( c_2Emin_2E_3D_3D_3E @ ( V1S_27 @ V2x )
@ ( c_2Ebool_2E_3F @ tyop_2Erealax_2Ereal
@ ^ [V3e: tyop_2Erealax_2Ereal] :
( c_2Ebool_2E_2F_5C @ ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V4y ) ) @ V3e ) @ ( V1S_27 @ V4y ) ) ) ) ) ) ) ) ).
thf(thm_2Emetric_2EMTOP__OPEN,axiom,
! [A_27a: $tType,V0S_27: A_27a > $o,V1m: tyop_2Emetric_2Emetric @ A_27a] :
( ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Emetric_2Emtop @ A_27a @ V1m ) @ V0S_27 )
<=> ! [V2x: A_27a] :
( ( V0S_27 @ V2x )
=> ? [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
& ! [V4y: A_27a] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V1m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V4y ) ) @ V3e )
=> ( V0S_27 @ V4y ) ) ) ) ) ).
thf(thm_2Emetric_2EBALL__OPEN,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Emetric_2Emtop @ A_27a @ V0m ) @ ( c_2Emetric_2EB @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) ) ) ).
thf(thm_2Emetric_2EBALL__NEIGH,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V2e )
=> ( c_2Etopology_2Eneigh @ A_27a @ ( c_2Emetric_2Emtop @ A_27a @ V0m ) @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ A_27a @ ( c_2Emetric_2EB @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ tyop_2Erealax_2Ereal @ V1x @ V2e ) ) @ V1x ) ) ) ).
thf(thm_2Emetric_2EMTOP__LIMPT,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2S_27: A_27a > $o] :
( ( c_2Etopology_2Elimpt @ A_27a @ ( c_2Emetric_2Emtop @ A_27a @ V0m ) @ V1x @ V2S_27 )
<=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4y: A_27a] :
( ( (~) @ ( V1x = V4y ) )
& ( V2S_27 @ V4y )
& ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V4y ) ) @ V3e ) ) ) ) ).
thf(thm_2Emetric_2EISMET__R1,axiom,
( c_2Emetric_2Eismet @ tyop_2Erealax_2Ereal
@ ( c_2Epair_2EUNCURRY @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal
@ ^ [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) ) ) ).
thf(thm_2Emetric_2EMR1__DEF,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Emetric_2Edist @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) )
= ( c_2Ereal_2Eabs @ ( c_2Ereal_2Ereal__sub @ V1y @ V0x ) ) ) ).
thf(thm_2Emetric_2EMR1__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal] :
( ( c_2Emetric_2Edist @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Erealax_2Ereal__add @ V0x @ V1d ) ) )
= ( c_2Ereal_2Eabs @ V1d ) ) ).
thf(thm_2Emetric_2EMR1__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal] :
( ( c_2Emetric_2Edist @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__sub @ V0x @ V1d ) ) )
= ( c_2Ereal_2Eabs @ V1d ) ) ).
thf(thm_2Emetric_2EMR1__ADD__POS,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1d )
=> ( ( c_2Emetric_2Edist @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Erealax_2Ereal__add @ V0x @ V1d ) ) )
= V1d ) ) ).
thf(thm_2Emetric_2EMR1__SUB__LE,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1d )
=> ( ( c_2Emetric_2Edist @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__sub @ V0x @ V1d ) ) )
= V1d ) ) ).
thf(thm_2Emetric_2EMR1__ADD__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1d )
=> ( ( c_2Emetric_2Edist @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Erealax_2Ereal__add @ V0x @ V1d ) ) )
= V1d ) ) ).
thf(thm_2Emetric_2EMR1__SUB__LT,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V1d )
=> ( ( c_2Emetric_2Edist @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__sub @ V0x @ V1d ) ) )
= V1d ) ) ).
thf(thm_2Emetric_2EMR1__BETWEEN1,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal,V2z: tyop_2Erealax_2Ereal] :
( ( ( c_2Erealax_2Ereal__lt @ V0x @ V2z )
& ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ V1y ) ) @ ( c_2Ereal_2Ereal__sub @ V2z @ V0x ) ) )
=> ( c_2Erealax_2Ereal__lt @ V1y @ V2z ) ) ).
thf(thm_2Emetric_2EMR1__LIMPT,axiom,
! [V0x: tyop_2Erealax_2Ereal] : ( c_2Etopology_2Elimpt @ tyop_2Erealax_2Ereal @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0x @ ( c_2Epred__set_2EUNIV @ tyop_2Erealax_2Ereal ) ) ).
%------------------------------------------------------------------------------