ITP001 Axioms: ITP121+5.ax


%------------------------------------------------------------------------------
% File     : ITP121+5 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 set theory export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : metric+2.ax [Gau20]
%          : HL4121+5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   42 (   4 unt;   0 def)
%            Number of atoms       :  169 (  27 equ)
%            Maximal formula atoms :   11 (   4 avg)
%            Number of connectives :  129 (   2   ~;   0   |;   9   &)
%                                         (   6 <=>; 112  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   7 avg)
%            Maximal term depth    :    9 (   2 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :   40 (  40 usr;  13 con; 0-5 aty)
%            Number of variables   :  116 ( 113   !;   3   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Emetric_2Emetric,axiom,
    ! [A0] :
      ( ne(A0)
     => ne(ty_2Emetric_2Emetric(A0)) ) ).

fof(mem_c_2Emetric_2EB,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Emetric_2EB(A_27a),arr(ty_2Emetric_2Emetric(A_27a),arr(ty_2Epair_2Eprod(A_27a,ty_2Erealax_2Ereal),arr(A_27a,bool)))) ) ).

fof(mem_c_2Emetric_2Edist,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Emetric_2Edist(A_27a),arr(ty_2Emetric_2Emetric(A_27a),arr(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Erealax_2Ereal))) ) ).

fof(mem_c_2Emetric_2Eismet,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Emetric_2Eismet(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Erealax_2Ereal),bool)) ) ).

fof(mem_c_2Emetric_2Emetric,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Emetric_2Emetric(A_27a),arr(arr(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Erealax_2Ereal),ty_2Emetric_2Emetric(A_27a))) ) ).

fof(mem_c_2Emetric_2Emr1,axiom,
    mem(c_2Emetric_2Emr1,ty_2Emetric_2Emetric(ty_2Erealax_2Ereal)) ).

fof(mem_c_2Emetric_2Emtop,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Emetric_2Emtop(A_27a),arr(ty_2Emetric_2Emetric(A_27a),ty_2Etopology_2Etopology(A_27a))) ) ).

fof(ax_thm_2Emetric_2Eismet,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,arr(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Erealax_2Ereal))
         => ( p(ap(c_2Emetric_2Eismet(A_27a),V0m))
          <=> ( ! [V1x] :
                  ( mem(V1x,A_27a)
                 => ! [V2y] :
                      ( mem(V2y,A_27a)
                     => ( ap(V0m,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
                      <=> V1x = V2y ) ) )
              & ! [V3x] :
                  ( mem(V3x,A_27a)
                 => ! [V4y] :
                      ( mem(V4y,A_27a)
                     => ! [V5z] :
                          ( mem(V5z,A_27a)
                         => p(ap(ap(c_2Ereal_2Ereal__lte,ap(V0m,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V4y),V5z))),ap(ap(c_2Erealax_2Ereal__add,ap(V0m,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3x),V4y))),ap(V0m,ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V3x),V5z))))) ) ) ) ) ) ) ) ).

fof(ax_thm_2Emetric_2Emetric__TY__DEF,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ? [V0rep] :
          ( mem(V0rep,arr(ty_2Emetric_2Emetric(A_27a),arr(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Erealax_2Ereal)))
          & p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(arr(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Erealax_2Ereal),ty_2Emetric_2Emetric(A_27a)),c_2Emetric_2Eismet(A_27a)),V0rep)) ) ) ).

fof(ax_thm_2Emetric_2Emetric__tybij,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ( ! [V0a] :
            ( mem(V0a,ty_2Emetric_2Emetric(A_27a))
           => ap(c_2Emetric_2Emetric(A_27a),ap(c_2Emetric_2Edist(A_27a),V0a)) = V0a )
        & ! [V1r] :
            ( mem(V1r,arr(ty_2Epair_2Eprod(A_27a,A_27a),ty_2Erealax_2Ereal))
           => ( p(ap(c_2Emetric_2Eismet(A_27a),V1r))
            <=> ap(c_2Emetric_2Edist(A_27a),ap(c_2Emetric_2Emetric(A_27a),V1r)) = V1r ) ) ) ) ).

fof(conj_thm_2Emetric_2EMETRIC__ISMET,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => p(ap(c_2Emetric_2Eismet(A_27a),ap(c_2Emetric_2Edist(A_27a),V0m))) ) ) ).

fof(conj_thm_2Emetric_2EMETRIC__ZERO,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2y] :
                  ( mem(V2y,A_27a)
                 => ( ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)
                  <=> V1x = V2y ) ) ) ) ) ).

fof(conj_thm_2Emetric_2EMETRIC__SAME,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V1x)) = ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0) ) ) ) ).

fof(conj_thm_2Emetric_2EMETRIC__POS,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2y] :
                  ( mem(V2y,A_27a)
                 => p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)))) ) ) ) ) ).

fof(conj_thm_2Emetric_2EMETRIC__SYM,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2y] :
                  ( mem(V2y,A_27a)
                 => ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)) = ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2y),V1x)) ) ) ) ) ).

fof(conj_thm_2Emetric_2EMETRIC__TRIANGLE,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2y] :
                  ( mem(V2y,A_27a)
                 => ! [V3z] :
                      ( mem(V3z,A_27a)
                     => p(ap(ap(c_2Ereal_2Ereal__lte,ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V3z))),ap(ap(c_2Erealax_2Ereal__add,ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y))),ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2y),V3z))))) ) ) ) ) ) ).

fof(conj_thm_2Emetric_2EMETRIC__NZ,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2y] :
                  ( mem(V2y,A_27a)
                 => ( V1x != V2y
                   => p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V2y)))) ) ) ) ) ) ).

fof(lameq_f2744,axiom,
    ! [A_27a,V2x] :
      ( mem(V2x,A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V3e] :
              ( mem(V3e,ty_2Erealax_2Ereal)
             => ! [V1S_27] :
                  ( mem(V1S_27,arr(A_27a,bool))
                 => ! [V4y] : ap(f2744(A_27a,V2x,V0m,V3e,V1S_27),V4y) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V4y))),V3e)),ap(V1S_27,V4y)) ) ) ) ) ).

fof(lameq_f2745,axiom,
    ! [A_27a,V2x] :
      ( mem(V2x,A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1S_27] :
              ( mem(V1S_27,arr(A_27a,bool))
             => ! [V3e] : ap(f2745(A_27a,V2x,V0m,V1S_27),V3e) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e)),ap(c_2Ebool_2E_21(A_27a),f2744(A_27a,V2x,V0m,V3e,V1S_27))) ) ) ) ).

fof(lameq_f2746,axiom,
    ! [A_27a,V0m] :
      ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
     => ! [V1S_27] :
          ( mem(V1S_27,arr(A_27a,bool))
         => ! [V2x] : ap(f2746(A_27a,V0m,V1S_27),V2x) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(V1S_27,V2x)),ap(c_2Ebool_2E_3F(ty_2Erealax_2Ereal),f2745(A_27a,V2x,V0m,V1S_27))) ) ) ).

fof(lameq_f2747,axiom,
    ! [A_27a,V0m] :
      ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
     => ! [V1S_27] : ap(f2747(A_27a,V0m),V1S_27) = ap(c_2Ebool_2E_21(A_27a),f2746(A_27a,V0m,V1S_27)) ) ).

fof(ax_thm_2Emetric_2Emtop,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ap(c_2Emetric_2Emtop(A_27a),V0m) = ap(c_2Etopology_2Etopology(A_27a),f2747(A_27a,V0m)) ) ) ).

fof(conj_thm_2Emetric_2Emtop__istopology,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => p(ap(c_2Etopology_2Eistopology(A_27a),f2747(A_27a,V0m))) ) ) ).

fof(conj_thm_2Emetric_2EMTOP__OPEN,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0S_27] :
          ( mem(V0S_27,arr(A_27a,bool))
         => ! [V1m] :
              ( mem(V1m,ty_2Emetric_2Emetric(A_27a))
             => ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(c_2Emetric_2Emtop(A_27a),V1m)),V0S_27))
              <=> ! [V2x] :
                    ( mem(V2x,A_27a)
                   => ( p(ap(V0S_27,V2x))
                     => ? [V3e] :
                          ( mem(V3e,ty_2Erealax_2Ereal)
                          & p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e))
                          & ! [V4y] :
                              ( mem(V4y,A_27a)
                             => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27a),V1m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V2x),V4y))),V3e))
                               => p(ap(V0S_27,V4y)) ) ) ) ) ) ) ) ) ) ).

fof(lameq_f2748,axiom,
    ! [A_27a,V1x] :
      ( mem(V1x,A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V2e] :
              ( mem(V2e,ty_2Erealax_2Ereal)
             => ! [V3y] : ap(f2748(A_27a,V1x,V0m,V2e),V3y) = ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V3y))),V2e) ) ) ) ).

fof(ax_thm_2Emetric_2Eball,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2e] :
                  ( mem(V2e,ty_2Erealax_2Ereal)
                 => ap(ap(c_2Emetric_2EB(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Erealax_2Ereal),V1x),V2e)) = f2748(A_27a,V1x,V0m,V2e) ) ) ) ) ).

fof(conj_thm_2Emetric_2EBALL__OPEN,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2e] :
                  ( mem(V2e,ty_2Erealax_2Ereal)
                 => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e))
                   => p(ap(ap(c_2Etopology_2Eopen__in(A_27a),ap(c_2Emetric_2Emtop(A_27a),V0m)),ap(ap(c_2Emetric_2EB(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Erealax_2Ereal),V1x),V2e)))) ) ) ) ) ) ).

fof(conj_thm_2Emetric_2EBALL__NEIGH,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2e] :
                  ( mem(V2e,ty_2Erealax_2Ereal)
                 => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V2e))
                   => p(ap(ap(c_2Etopology_2Eneigh(A_27a),ap(c_2Emetric_2Emtop(A_27a),V0m)),ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),A_27a),ap(ap(c_2Emetric_2EB(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Erealax_2Ereal),V1x),V2e))),V1x))) ) ) ) ) ) ).

fof(conj_thm_2Emetric_2EMTOP__LIMPT,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0m] :
          ( mem(V0m,ty_2Emetric_2Emetric(A_27a))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2S_27] :
                  ( mem(V2S_27,arr(A_27a,bool))
                 => ( p(ap(ap(ap(c_2Etopology_2Elimpt(A_27a),ap(c_2Emetric_2Emtop(A_27a),V0m)),V1x),V2S_27))
                  <=> ! [V3e] :
                        ( mem(V3e,ty_2Erealax_2Ereal)
                       => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V3e))
                         => ? [V4y] :
                              ( mem(V4y,A_27a)
                              & V1x != V4y
                              & p(ap(V2S_27,V4y))
                              & p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(A_27a),V0m),ap(ap(c_2Epair_2E_2C(A_27a,A_27a),V1x),V4y))),V3e)) ) ) ) ) ) ) ) ) ).

fof(lameq_f2749,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1y] : ap(f2749(V0x),V1y) = ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1y),V0x)) ) ).

fof(lameq_f2750,axiom,
    ! [V0x] : ap(f2750,V0x) = f2749(V0x) ).

fof(conj_thm_2Emetric_2EISMET__R1,axiom,
    p(ap(c_2Emetric_2Eismet(ty_2Erealax_2Ereal),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f2750))) ).

fof(ax_thm_2Emetric_2Emr1,axiom,
    c_2Emetric_2Emr1 = ap(c_2Emetric_2Emetric(ty_2Erealax_2Ereal),ap(c_2Epair_2EUNCURRY(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),f2750)) ).

fof(conj_thm_2Emetric_2EMR1__DEF,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1y] :
          ( mem(V1y,ty_2Erealax_2Ereal)
         => ap(ap(c_2Emetric_2Edist(ty_2Erealax_2Ereal),c_2Emetric_2Emr1),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y)) = ap(c_2Ereal_2Eabs,ap(ap(c_2Ereal_2Ereal__sub,V1y),V0x)) ) ) ).

fof(conj_thm_2Emetric_2EMR1__ADD,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1d] :
          ( mem(V1d,ty_2Erealax_2Ereal)
         => ap(ap(c_2Emetric_2Edist(ty_2Erealax_2Ereal),c_2Emetric_2Emr1),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(ap(c_2Erealax_2Ereal__add,V0x),V1d))) = ap(c_2Ereal_2Eabs,V1d) ) ) ).

fof(conj_thm_2Emetric_2EMR1__SUB,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1d] :
          ( mem(V1d,ty_2Erealax_2Ereal)
         => ap(ap(c_2Emetric_2Edist(ty_2Erealax_2Ereal),c_2Emetric_2Emr1),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1d))) = ap(c_2Ereal_2Eabs,V1d) ) ) ).

fof(conj_thm_2Emetric_2EMR1__ADD__POS,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1d] :
          ( mem(V1d,ty_2Erealax_2Ereal)
         => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1d))
           => ap(ap(c_2Emetric_2Edist(ty_2Erealax_2Ereal),c_2Emetric_2Emr1),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(ap(c_2Erealax_2Ereal__add,V0x),V1d))) = V1d ) ) ) ).

fof(conj_thm_2Emetric_2EMR1__SUB__LE,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1d] :
          ( mem(V1d,ty_2Erealax_2Ereal)
         => ( p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1d))
           => ap(ap(c_2Emetric_2Edist(ty_2Erealax_2Ereal),c_2Emetric_2Emr1),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1d))) = V1d ) ) ) ).

fof(conj_thm_2Emetric_2EMR1__ADD__LT,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1d] :
          ( mem(V1d,ty_2Erealax_2Ereal)
         => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1d))
           => ap(ap(c_2Emetric_2Edist(ty_2Erealax_2Ereal),c_2Emetric_2Emr1),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(ap(c_2Erealax_2Ereal__add,V0x),V1d))) = V1d ) ) ) ).

fof(conj_thm_2Emetric_2EMR1__SUB__LT,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1d] :
          ( mem(V1d,ty_2Erealax_2Ereal)
         => ( p(ap(ap(c_2Erealax_2Ereal__lt,ap(c_2Ereal_2Ereal__of__num,c_2Enum_2E0)),V1d))
           => ap(ap(c_2Emetric_2Edist(ty_2Erealax_2Ereal),c_2Emetric_2Emr1),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),ap(ap(c_2Ereal_2Ereal__sub,V0x),V1d))) = V1d ) ) ) ).

fof(conj_thm_2Emetric_2EMR1__BETWEEN1,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => ! [V1y] :
          ( mem(V1y,ty_2Erealax_2Ereal)
         => ! [V2z] :
              ( mem(V2z,ty_2Erealax_2Ereal)
             => ( ( p(ap(ap(c_2Erealax_2Ereal__lt,V0x),V2z))
                  & p(ap(ap(c_2Erealax_2Ereal__lt,ap(ap(c_2Emetric_2Edist(ty_2Erealax_2Ereal),c_2Emetric_2Emr1),ap(ap(c_2Epair_2E_2C(ty_2Erealax_2Ereal,ty_2Erealax_2Ereal),V0x),V1y))),ap(ap(c_2Ereal_2Ereal__sub,V2z),V0x))) )
               => p(ap(ap(c_2Erealax_2Ereal__lt,V1y),V2z)) ) ) ) ) ).

fof(conj_thm_2Emetric_2EMR1__LIMPT,axiom,
    ! [V0x] :
      ( mem(V0x,ty_2Erealax_2Ereal)
     => p(ap(ap(ap(c_2Etopology_2Elimpt(ty_2Erealax_2Ereal),ap(c_2Emetric_2Emtop(ty_2Erealax_2Ereal),c_2Emetric_2Emr1)),V0x),c_2Epred__set_2EUNIV(ty_2Erealax_2Ereal))) ) ).

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