ITP001 Axioms: ITP121^7.ax


%------------------------------------------------------------------------------
% File     : ITP121^7 : TPTP v9.0.0. 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 ) ) ).

%------------------------------------------------------------------------------