ITP001 Axioms: ITP121+5.ax
%------------------------------------------------------------------------------
% File : ITP121+5 : TPTP v8.2.0. 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))) ) ).
%------------------------------------------------------------------------------