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 : 47 ( 5 unt; 10 typ; 0 def)
% Number of atoms : 674 ( 22 equ; 0 cnn)
% Maximal formula atoms : 44 ( 14 avg)
% Number of connectives : 1050 ( 2 ~; 0 |; 8 &; 980 @)
% ( 6 <=>; 54 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 11 avg; 980 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 43 usr; 35 con; 0-2 aty)
% Number of variables : 105 ( 13 ^ 89 !; 3 ?; 105 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Emetric_2Emetric,type,
ty_2Emetric_2Emetric: del > del ).
thf(tp_c_2Emetric_2EB,type,
c_2Emetric_2EB: del > $i ).
thf(mem_c_2Emetric_2EB,axiom,
! [A_27a: del] : ( 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 ) ) ) ) ).
thf(tp_c_2Emetric_2Edist,type,
c_2Emetric_2Edist: del > $i ).
thf(mem_c_2Emetric_2Edist,axiom,
! [A_27a: del] : ( mem @ ( c_2Emetric_2Edist @ A_27a ) @ ( arr @ ( ty_2Emetric_2Emetric @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) ) ) ).
thf(tp_c_2Emetric_2Eismet,type,
c_2Emetric_2Eismet: del > $i ).
thf(mem_c_2Emetric_2Eismet,axiom,
! [A_27a: del] : ( mem @ ( c_2Emetric_2Eismet @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) @ bool ) ) ).
thf(tp_c_2Emetric_2Emetric,type,
c_2Emetric_2Emetric: del > $i ).
thf(mem_c_2Emetric_2Emetric,axiom,
! [A_27a: del] : ( mem @ ( c_2Emetric_2Emetric @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) @ ( ty_2Emetric_2Emetric @ A_27a ) ) ) ).
thf(stp_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,type,
tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal: $tType ).
thf(stp_inj_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,type,
inj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal: tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal > $i ).
thf(stp_surj_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,type,
surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal: $i > tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal ).
thf(stp_inj_surj_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal] :
( ( surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ ( inj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,axiom,
! [X: tp__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal] : ( mem @ ( inj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ X ) @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) ) ).
thf(stp_iso_mem_c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) )
=> ( X
= ( inj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ ( surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ X ) ) ) ) ).
thf(tp_c_2Emetric_2Emr1,type,
c_2Emetric_2Emr1: $i ).
thf(mem_c_2Emetric_2Emr1,axiom,
mem @ c_2Emetric_2Emr1 @ ( ty_2Emetric_2Emetric @ ty_2Erealax_2Ereal ) ).
thf(tp_c_2Emetric_2Emtop,type,
c_2Emetric_2Emtop: del > $i ).
thf(mem_c_2Emetric_2Emtop,axiom,
! [A_27a: del] : ( mem @ ( c_2Emetric_2Emtop @ A_27a ) @ ( arr @ ( ty_2Emetric_2Emetric @ A_27a ) @ ( ty_2Etopology_2Etopology @ A_27a ) ) ) ).
thf(ax_thm_2Emetric_2Eismet,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ty_2Erealax_2Ereal ) )
=> ( ( p @ ( ap @ ( c_2Emetric_2Eismet @ A_27a ) @ V0m ) )
<=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ V0m @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V1x = V2y ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emetric_2Emetric__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( 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 ) ) ) ).
thf(ax_thm_2Emetric_2Emetric__tybij,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ( ( ap @ ( c_2Emetric_2Emetric @ A_27a ) @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( 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 ) ) ) ) ).
thf(conj_thm_2Emetric_2EMETRIC__ISMET,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ( p @ ( ap @ ( c_2Emetric_2Eismet @ A_27a ) @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) ) ) ) ).
thf(conj_thm_2Emetric_2EMETRIC__ZERO,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
<=> ( V1x = V2y ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMETRIC__SAME,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V1x ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMETRIC__POS,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMETRIC__SYM,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMETRIC__TRIANGLE,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( 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 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMETRIC__NZ,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( V1x != V2y )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Emetric_2Edist @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emetric_2Emtop,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ( ( ap @ ( c_2Emetric_2Emtop @ A_27a ) @ V0m )
= ( ap @ ( c_2Etopology_2Etopology @ A_27a )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1S_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1S_27 @ V2x ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3e: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V3e ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2Emtop__istopology,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ( p
@ ( ap @ ( c_2Etopology_2Eistopology @ A_27a )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1S_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V1S_27 @ V2x ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V3e: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V3e ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMTOP__OPEN,axiom,
! [A_27a: del,V0S_27: $i] :
( ( mem @ V0S_27 @ ( arr @ A_27a @ bool ) )
=> ! [V1m: $i] :
( ( 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: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0S_27 @ V2x ) )
=> ? [V3e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
& ! [V4y: $i] :
( ( 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 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ( p @ ( ap @ V0S_27 @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Emetric_2Eball,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( ap @ ( ap @ ( c_2Emetric_2EB @ A_27a ) @ V0m ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ty_2Erealax_2Ereal ) @ V1x ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) )
= ( lam @ A_27a
@ ^ [V3y: $i] : ( 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 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EBALL__OPEN,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ 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 ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EBALL__NEIGH,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2e: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ 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 ) @ ( inj__ty_2Erealax_2Ereal @ V2e ) ) ) ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMTOP__LIMPT,axiom,
! [A_27a: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Emetric_2Emetric @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2S_27: $i] :
( ( 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: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) )
=> ? [V4y: $i] :
( ( 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 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V3e ) ) ) ) ) ) ) ) ) ).
thf(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 )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V1y: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V1y ) @ V0x ) ) ) ) ) ) ) ).
thf(ax_thm_2Emetric_2Emr1,axiom,
( ( surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal @ c_2Emetric_2Emr1 )
= ( surj__c_ty_2Emetric_2Emetric_ty_2Erealax_2Ereal
@ ( ap @ ( c_2Emetric_2Emetric @ ty_2Erealax_2Ereal )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal @ ty_2Erealax_2Ereal )
@ ( lam @ ty_2Erealax_2Ereal
@ ^ [V0x: $i] :
( lam @ ty_2Erealax_2Ereal
@ ^ [V1y: $i] : ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ V1y ) @ V0x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMR1__DEF,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal] :
( ( surj__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 ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) ) ).
thf(conj_thm_2Emetric_2EMR1__ADD,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal] :
( ( surj__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 ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) ).
thf(conj_thm_2Emetric_2EMR1__SUB,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal] :
( ( surj__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 ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
= ( surj__ty_2Erealax_2Ereal @ ( ap @ c_2Ereal_2Eabs @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) ).
thf(conj_thm_2Emetric_2EMR1__ADD__POS,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) )
=> ( ( surj__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 ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
= V1d ) ) ).
thf(conj_thm_2Emetric_2EMR1__SUB__LE,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Ereal_2Ereal__lte @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) )
=> ( ( surj__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 ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
= V1d ) ) ).
thf(conj_thm_2Emetric_2EMR1__ADD__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) )
=> ( ( surj__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 ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Erealax_2Ereal__add @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
= V1d ) ) ).
thf(conj_thm_2Emetric_2EMR1__SUB__LT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1d: tp__ty_2Erealax_2Ereal] :
( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( ap @ c_2Ereal_2Ereal__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) )
=> ( ( surj__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 ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1d ) ) ) ) )
= V1d ) ) ).
thf(conj_thm_2Emetric_2EMR1__BETWEEN1,axiom,
! [V0x: tp__ty_2Erealax_2Ereal,V1y: tp__ty_2Erealax_2Ereal,V2z: tp__ty_2Erealax_2Ereal] :
( ( ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ 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 ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) ) ) @ ( ap @ ( ap @ c_2Ereal_2Ereal__sub @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) ) ) )
=> ( p @ ( ap @ ( ap @ c_2Erealax_2Ereal__lt @ ( inj__ty_2Erealax_2Ereal @ V1y ) ) @ ( inj__ty_2Erealax_2Ereal @ V2z ) ) ) ) ).
thf(conj_thm_2Emetric_2EMR1__LIMPT,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etopology_2Elimpt @ ty_2Erealax_2Ereal ) @ ( ap @ ( c_2Emetric_2Emtop @ ty_2Erealax_2Ereal ) @ c_2Emetric_2Emr1 ) ) @ ( inj__ty_2Erealax_2Ereal @ V0x ) ) @ ( c_2Epred__set_2EUNIV @ ty_2Erealax_2Ereal ) ) ) ).
%------------------------------------------------------------------------------