ITP001 Axioms: ITP124^7.ax
%------------------------------------------------------------------------------
% File : ITP124^7 : 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 : nets.ax [Gau19]
% : HL4124^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 72 ( 8 unt; 37 typ; 0 def)
% Number of atoms : 112 ( 9 equ; 6 cnn)
% Maximal formula atoms : 9 ( 1 avg)
% Number of connectives : 1009 ( 6 ~; 1 |; 33 &; 905 @)
% ( 16 <=>; 48 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 13 avg; 905 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 235 ( 235 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 34 usr; 3 con; 0-5 aty)
% Number of variables : 209 ( 12 ^ 164 !; 17 ?; 209 :)
% ( 16 !>; 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_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_2Earithmetic_2E_3E_3D,type,
c_2Earithmetic_2E_3E_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
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_2Enets_2Ebounded,type,
c_2Enets_2Ebounded:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ ( tyop_2Emetric_2Emetric @ A_27a ) @ ( A_27b > A_27b > $o ) ) > ( A_27b > A_27a ) > $o ) ).
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_2Enets_2Edorder,type,
c_2Enets_2Edorder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_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_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_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
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_2Enets_2Etends,type,
c_2Enets_2Etends:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a ) > A_27a > ( tyop_2Epair_2Eprod @ ( tyop_2Etopology_2Etopology @ A_27a ) @ ( A_27b > A_27b > $o ) ) > $o ) ).
thf(c_2Enets_2Etendsto,type,
c_2Enets_2Etendsto:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( tyop_2Emetric_2Emetric @ A_27a ) @ A_27a ) > A_27a > A_27a > $o ) ).
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_2Enets_2Edorder,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( V0g @ V1x @ V1x )
& ( V0g @ V2y @ V2y ) )
=> ? [V3z: A_27a] :
( ( V0g @ V3z @ V3z )
& ! [V4w: A_27a] :
( ( V0g @ V4w @ V3z )
=> ( ( V0g @ V4w @ V1x )
& ( V0g @ V4w @ V2y ) ) ) ) ) ) ).
thf(thm_2Enets_2Etends,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > A_27a,V1l: A_27a,V2top: tyop_2Etopology_2Etopology @ A_27a,V3g: A_27b > A_27b > $o] :
( ( c_2Enets_2Etends @ A_27a @ A_27b @ V0s @ V1l @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ A_27a ) @ ( A_27b > A_27b > $o ) @ V2top @ V3g ) )
<=> ! [V4N: A_27a > $o] :
( ( c_2Etopology_2Eneigh @ A_27a @ V2top @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ A_27a @ V4N @ V1l ) )
=> ? [V5n: A_27b] :
( ( V3g @ V5n @ V5n )
& ! [V6m: A_27b] :
( ( V3g @ V6m @ V5n )
=> ( V4N @ ( V0s @ V6m ) ) ) ) ) ) ).
thf(thm_2Enets_2Ebounded,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1g: A_27b > A_27b > $o,V2f: A_27b > A_27a] :
( ( c_2Enets_2Ebounded @ A_27a @ A_27b @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ A_27a ) @ ( A_27b > A_27b > $o ) @ V0m @ V1g ) @ V2f )
<=> ? [V3k: tyop_2Erealax_2Ereal,V4x: A_27a,V5N: A_27b] :
( ( V1g @ V5N @ V5N )
& ! [V6n: A_27b] :
( ( V1g @ V6n @ V5N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V0m @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ ( V2f @ V6n ) @ V4x ) ) @ V3k ) ) ) ) ).
thf(thm_2Enets_2Etendsto,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( c_2Enets_2Etendsto @ A_27a @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ A_27a ) @ A_27a @ V0m @ V1x ) @ V2y @ V3z )
<=> ( ( 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 ) ) )
& ( c_2Ereal_2Ereal__lte @ ( 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 @ V1x @ V3z ) ) ) ) ) ).
thf(thm_2Enets_2EDORDER__LEMMA,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1P: A_27a > $o,V2Q: A_27a > $o] :
( ( ? [V3n: A_27a] :
( ( V0g @ V3n @ V3n )
& ! [V4m: A_27a] :
( ( V0g @ V4m @ V3n )
=> ( V1P @ V4m ) ) )
& ? [V5n: A_27a] :
( ( V0g @ V5n @ V5n )
& ! [V6m: A_27a] :
( ( V0g @ V6m @ V5n )
=> ( V2Q @ V6m ) ) ) )
=> ? [V7n: A_27a] :
( ( V0g @ V7n @ V7n )
& ! [V8m: A_27a] :
( ( V0g @ V8m @ V7n )
=> ( ( V1P @ V8m )
& ( V2Q @ V8m ) ) ) ) ) ) ).
thf(thm_2Enets_2EDORDER__NGE,axiom,
c_2Enets_2Edorder @ tyop_2Enum_2Enum @ c_2Earithmetic_2E_3E_3D ).
thf(thm_2Enets_2EDORDER__TENDSTO,axiom,
! [A_27a: $tType,V0m: tyop_2Emetric_2Emetric @ A_27a,V1x: A_27a] : ( c_2Enets_2Edorder @ A_27a @ ( c_2Enets_2Etendsto @ A_27a @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ A_27a ) @ A_27a @ V0m @ V1x ) ) ) ).
thf(thm_2Enets_2EMTOP__TENDS,axiom,
! [A_27a: $tType,A_27b: $tType,V0d: tyop_2Emetric_2Emetric @ A_27a,V1g: A_27b > A_27b > $o,V2x: A_27b > A_27a,V3x0: A_27a] :
( ( c_2Enets_2Etends @ A_27a @ A_27b @ V2x @ V3x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ A_27a ) @ ( A_27b > A_27b > $o ) @ ( c_2Emetric_2Emtop @ A_27a @ V0d ) @ V1g ) )
<=> ! [V4e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V4e )
=> ? [V5n: A_27b] :
( ( V1g @ V5n @ V5n )
& ! [V6m: A_27b] :
( ( V1g @ V6m @ V5n )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V0d @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ ( V2x @ V6m ) @ V3x0 ) ) @ V4e ) ) ) ) ) ).
thf(thm_2Enets_2EMTOP__TENDS__UNIQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0x1: A_27a,V1x0: A_27a,V2x: A_27b > A_27a,V3g: A_27b > A_27b > $o,V4d: tyop_2Emetric_2Emetric @ A_27a] :
( ( c_2Enets_2Edorder @ A_27b @ V3g )
=> ( ( ( c_2Enets_2Etends @ A_27a @ A_27b @ V2x @ V1x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ A_27a ) @ ( A_27b > A_27b > $o ) @ ( c_2Emetric_2Emtop @ A_27a @ V4d ) @ V3g ) )
& ( c_2Enets_2Etends @ A_27a @ A_27b @ V2x @ V0x1 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ A_27a ) @ ( A_27b > A_27b > $o ) @ ( c_2Emetric_2Emtop @ A_27a @ V4d ) @ V3g ) ) )
=> ( V1x0 = V0x1 ) ) ) ).
thf(thm_2Enets_2ESEQ__TENDS,axiom,
! [A_27a: $tType,V0d: tyop_2Emetric_2Emetric @ A_27a,V1x: tyop_2Enum_2Enum > A_27a,V2x0: A_27a] :
( ( c_2Enets_2Etends @ A_27a @ tyop_2Enum_2Enum @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ A_27a ) @ ( tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ) @ ( c_2Emetric_2Emtop @ A_27a @ V0d ) @ c_2Earithmetic_2E_3E_3D ) )
<=> ! [V3e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V3e )
=> ? [V4N: tyop_2Enum_2Enum] :
! [V5n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3E_3D @ V5n @ V4N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V0d @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ ( V1x @ V5n ) @ V2x0 ) ) @ V3e ) ) ) ) ).
thf(thm_2Enets_2ELIM__TENDS,axiom,
! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Emetric_2Emetric @ A_27a,V1m2: tyop_2Emetric_2Emetric @ A_27b,V2f: A_27a > A_27b,V3x0: A_27a,V4y0: A_27b] :
( ( c_2Etopology_2Elimpt @ A_27a @ ( c_2Emetric_2Emtop @ A_27a @ V0m1 ) @ V3x0 @ ( c_2Epred__set_2EUNIV @ A_27a ) )
=> ( ( c_2Enets_2Etends @ A_27b @ A_27a @ V2f @ V4y0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ A_27b ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ A_27b @ V1m2 ) @ ( c_2Enets_2Etendsto @ A_27a @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ A_27a ) @ A_27a @ V0m1 @ V3x0 ) ) ) )
<=> ! [V5e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e )
=> ? [V6d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V6d )
& ! [V7x: A_27a] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Emetric_2Edist @ A_27a @ V0m1 @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V3x0 ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Emetric_2Edist @ A_27a @ V0m1 @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V3x0 ) ) @ V6d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27b @ V1m2 @ ( c_2Epair_2E_2C @ A_27b @ A_27b @ ( V2f @ V7x ) @ V4y0 ) ) @ V5e ) ) ) ) ) ) ).
thf(thm_2Enets_2ELIM__TENDS2,axiom,
! [A_27a: $tType,A_27b: $tType,V0m1: tyop_2Emetric_2Emetric @ A_27a,V1m2: tyop_2Emetric_2Emetric @ A_27b,V2f: A_27a > A_27b,V3x0: A_27a,V4y0: A_27b] :
( ( c_2Etopology_2Elimpt @ A_27a @ ( c_2Emetric_2Emtop @ A_27a @ V0m1 ) @ V3x0 @ ( c_2Epred__set_2EUNIV @ A_27a ) )
=> ( ( c_2Enets_2Etends @ A_27b @ A_27a @ V2f @ V4y0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ A_27b ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ A_27b @ V1m2 ) @ ( c_2Enets_2Etendsto @ A_27a @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ A_27a ) @ A_27a @ V0m1 @ V3x0 ) ) ) )
<=> ! [V5e: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V5e )
=> ? [V6d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V6d )
& ! [V7x: A_27a] :
( ( ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Emetric_2Edist @ A_27a @ V0m1 @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V3x0 ) ) )
& ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27a @ V0m1 @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V3x0 ) ) @ V6d ) )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Emetric_2Edist @ A_27b @ V1m2 @ ( c_2Epair_2E_2C @ A_27b @ A_27b @ ( V2f @ V7x ) @ V4y0 ) ) @ V5e ) ) ) ) ) ) ).
thf(thm_2Enets_2EMR1__BOUNDED,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o,V1f: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ A_27a @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ c_2Emetric_2Emr1 @ V0g ) @ V1f )
<=> ? [V2k: tyop_2Erealax_2Ereal,V3N: A_27a] :
( ( V0g @ V3N @ V3N )
& ! [V4n: A_27a] :
( ( V0g @ V4n @ V3N )
=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Eabs @ ( V1f @ V4n ) ) @ V2k ) ) ) ) ).
thf(thm_2Enets_2ENET__NULL,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o,V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal] :
( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
= ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V3n: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V1x @ V3n ) @ V2x0 )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ).
thf(thm_2Enets_2ENET__CONV__BOUNDED,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o,V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal] :
( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
=> ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ A_27a @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ c_2Emetric_2Emr1 @ V0g ) @ V1x ) ) ).
thf(thm_2Enets_2ENET__CONV__NZ,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o,V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( (~)
@ ( V2x0
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ? [V3N: A_27a] :
( ( V0g @ V3N @ V3N )
& ! [V4n: A_27a] :
( ( V0g @ V4n @ V3N )
=> ( (~)
@ ( ( V1x @ V4n )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ) ) ) ).
thf(thm_2Enets_2ENET__CONV__IBOUNDED,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o,V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( (~)
@ ( V2x0
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ A_27a @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ c_2Emetric_2Emr1 @ V0g )
@ ^ [V3n: A_27a] : ( c_2Erealax_2Einv @ ( V1x @ V3n ) ) ) ) ).
thf(thm_2Enets_2ENET__NULL__ADD,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2y: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V2y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V3n: A_27a] : ( c_2Erealax_2Ereal__add @ ( V1x @ V3n ) @ ( V2y @ V3n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ) ).
thf(thm_2Enets_2ENET__NULL__MUL,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2y: A_27a > tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Ebounded @ tyop_2Erealax_2Ereal @ A_27a @ ( c_2Epair_2E_2C @ ( tyop_2Emetric_2Emetric @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ c_2Emetric_2Emr1 @ V0g ) @ V1x )
& ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V2y @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V3n: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V1x @ V3n ) @ ( V2y @ V3n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ) ).
thf(thm_2Enets_2ENET__NULL__CMUL,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o,V1k: tyop_2Erealax_2Ereal,V2x: A_27a > tyop_2Erealax_2Ereal] :
( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V2x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V3n: A_27a] : ( c_2Erealax_2Ereal__mul @ V1k @ ( V2x @ V3n ) )
@ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ).
thf(thm_2Enets_2ENET__ADD,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal,V3y: A_27a > tyop_2Erealax_2Ereal,V4y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V3y @ V4y0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V5n: A_27a] : ( c_2Erealax_2Ereal__add @ ( V1x @ V5n ) @ ( V3y @ V5n ) )
@ ( c_2Erealax_2Ereal__add @ V2x0 @ V4y0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ) ).
thf(thm_2Enets_2ENET__NEG,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal] :
( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
= ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V3n: A_27a] : ( c_2Erealax_2Ereal__neg @ ( V1x @ V3n ) )
@ ( c_2Erealax_2Ereal__neg @ V2x0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ) ).
thf(thm_2Enets_2ENET__SUB,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal,V3y: A_27a > tyop_2Erealax_2Ereal,V4y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V3y @ V4y0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V5n: A_27a] : ( c_2Ereal_2Ereal__sub @ ( V1x @ V5n ) @ ( V3y @ V5n ) )
@ ( c_2Ereal_2Ereal__sub @ V2x0 @ V4y0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ) ).
thf(thm_2Enets_2ENET__MUL,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2y: A_27a > tyop_2Erealax_2Ereal,V3x0: tyop_2Erealax_2Ereal,V4y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V3x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V2y @ V4y0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V5n: A_27a] : ( c_2Erealax_2Ereal__mul @ ( V1x @ V5n ) @ ( V2y @ V5n ) )
@ ( c_2Erealax_2Ereal__mul @ V3x0 @ V4y0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ) ).
thf(thm_2Enets_2ENET__INV,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( (~)
@ ( V2x0
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V3n: A_27a] : ( c_2Erealax_2Einv @ ( V1x @ V3n ) )
@ ( c_2Erealax_2Einv @ V2x0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ) ).
thf(thm_2Enets_2ENET__DIV,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal,V3y: A_27a > tyop_2Erealax_2Ereal,V4y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V3y @ V4y0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( (~)
@ ( V4y0
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V5n: A_27a] : ( c_2Ereal_2E_2F @ ( V1x @ V5n ) @ ( V3y @ V5n ) )
@ ( c_2Ereal_2E_2F @ V2x0 @ V4y0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ) ).
thf(thm_2Enets_2ENET__ABS,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o,V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal] :
( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
=> ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a
@ ^ [V3n: A_27a] : ( c_2Ereal_2Eabs @ ( V1x @ V3n ) )
@ ( c_2Ereal_2Eabs @ V2x0 )
@ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) ) ) ).
thf(thm_2Enets_2ENET__LE,axiom,
! [A_27a: $tType,V0g: A_27a > A_27a > $o] :
( ( c_2Enets_2Edorder @ A_27a @ V0g )
=> ! [V1x: A_27a > tyop_2Erealax_2Ereal,V2x0: tyop_2Erealax_2Ereal,V3y: A_27a > tyop_2Erealax_2Ereal,V4y0: tyop_2Erealax_2Ereal] :
( ( ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V1x @ V2x0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ( c_2Enets_2Etends @ tyop_2Erealax_2Ereal @ A_27a @ V3y @ V4y0 @ ( c_2Epair_2E_2C @ ( tyop_2Etopology_2Etopology @ tyop_2Erealax_2Ereal ) @ ( A_27a > A_27a > $o ) @ ( c_2Emetric_2Emtop @ tyop_2Erealax_2Ereal @ c_2Emetric_2Emr1 ) @ V0g ) )
& ? [V5N: A_27a] :
( ( V0g @ V5N @ V5N )
& ! [V6n: A_27a] :
( ( V0g @ V6n @ V5N )
=> ( c_2Ereal_2Ereal__lte @ ( V1x @ V6n ) @ ( V3y @ V6n ) ) ) ) )
=> ( c_2Ereal_2Ereal__lte @ V2x0 @ V4y0 ) ) ) ).
%------------------------------------------------------------------------------