SET007 Axioms: SET007+265.ax
%------------------------------------------------------------------------------
% File : SET007+265 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Submetric Spaces - Part I
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : sub_metr [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 75 ( 39 unt; 0 def)
% Number of atoms : 268 ( 19 equ)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 233 ( 40 ~; 0 |; 102 &)
% ( 8 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-4 aty)
% Number of functors : 26 ( 26 usr; 6 con; 0-5 aty)
% Number of variables : 87 ( 84 !; 3 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_sub_metr,axiom,
( ~ v3_struct_0(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
& v1_metric_1(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
& v6_metric_1(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
& v8_metric_1(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
& v9_metric_1(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
& v2_sub_metr(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1)) ) ).
fof(rc1_sub_metr,axiom,
? [A] :
( l1_metric_1(A)
& ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& v2_sub_metr(A) ) ).
fof(rc2_sub_metr,axiom,
? [A] :
( l1_metric_1(A)
& ~ v3_struct_0(A)
& v1_metric_1(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v2_sub_metr(A)
& v3_sub_metr(A) ) ).
fof(cc1_sub_metr,axiom,
! [A] :
( l1_metric_1(A)
=> ( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A) )
=> ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& v2_sub_metr(A) ) ) ) ).
fof(cc2_sub_metr,axiom,
! [A] :
( l1_metric_1(A)
=> ( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v2_sub_metr(A)
& v3_sub_metr(A) )
=> ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& v2_sub_metr(A)
& v3_sub_metr(A) ) ) ) ).
fof(fc2_sub_metr,axiom,
( ~ v3_struct_0(k2_sub_metr)
& v1_metric_1(k2_sub_metr) ) ).
fof(fc3_sub_metr,axiom,
( ~ v3_struct_0(k2_sub_metr)
& v1_metric_1(k2_sub_metr)
& v6_metric_1(k2_sub_metr)
& v8_metric_1(k2_sub_metr)
& v9_metric_1(k2_sub_metr) ) ).
fof(t1_sub_metr,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> r1_xreal_0(k2_square_1(A,B),k2_xcmplx_0(A,B)) ) ) ) ).
fof(t2_sub_metr,axiom,
! [A] :
( ( v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& r1_xreal_0(k4_metric_1(A,B,C),np__0) ) ) ) ) ).
fof(t3_sub_metr,axiom,
$true ).
fof(t4_sub_metr,axiom,
! [A] :
( m1_subset_1(A,k1_tarski(k1_xboole_0))
=> k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,A) = np__0 ) ).
fof(t5_sub_metr,axiom,
! [A] :
( m1_subset_1(A,k1_tarski(k1_xboole_0))
=> ! [B] :
( m1_subset_1(B,k1_tarski(k1_xboole_0))
=> ~ ( A != B
& r1_xreal_0(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,B),np__0) ) ) ) ).
fof(t6_sub_metr,axiom,
! [A] :
( m1_subset_1(A,k1_tarski(k1_xboole_0))
=> ! [B] :
( m1_subset_1(B,k1_tarski(k1_xboole_0))
=> k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,B) = k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,B,A) ) ) ).
fof(t7_sub_metr,axiom,
! [A] :
( m1_subset_1(A,k1_tarski(k1_xboole_0))
=> ! [B] :
( m1_subset_1(B,k1_tarski(k1_xboole_0))
=> ! [C] :
( m1_subset_1(C,k1_tarski(k1_xboole_0))
=> r1_xreal_0(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,C),k2_xcmplx_0(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,B),k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,B,C))) ) ) ) ).
fof(t8_sub_metr,axiom,
! [A] :
( m1_subset_1(A,k1_tarski(k1_xboole_0))
=> ! [B] :
( m1_subset_1(B,k1_tarski(k1_xboole_0))
=> ! [C] :
( m1_subset_1(C,k1_tarski(k1_xboole_0))
=> r1_xreal_0(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,C),k4_square_1(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,B),k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,B,C))) ) ) ) ).
fof(d1_sub_metr,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( v1_sub_metr(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ~ ( C != D
& r1_xreal_0(k1_metric_1(A,A,B,C,D),np__0) ) ) ) ) ) ) ).
fof(d2_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v2_sub_metr(A)
<=> v1_sub_metr(u1_metric_1(A),u1_struct_0(A)) ) ) ).
fof(t9_sub_metr,axiom,
$true ).
fof(t10_sub_metr,axiom,
$true ).
fof(t11_sub_metr,axiom,
$true ).
fof(t12_sub_metr,axiom,
$true ).
fof(t13_sub_metr,axiom,
$true ).
fof(t14_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != C
& r1_xreal_0(k2_metric_1(A,B,C),np__0) ) ) )
<=> v2_sub_metr(A) ) ) ).
fof(t15_sub_metr,axiom,
$true ).
fof(t16_sub_metr,axiom,
$true ).
fof(t17_sub_metr,axiom,
$true ).
fof(t18_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v2_sub_metr(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_xreal_0(np__0,k2_metric_1(A,B,C)) ) ) ) ).
fof(t19_sub_metr,axiom,
$true ).
fof(t20_sub_metr,axiom,
$true ).
fof(t21_sub_metr,axiom,
$true ).
fof(t22_sub_metr,axiom,
$true ).
fof(t23_sub_metr,axiom,
$true ).
fof(t24_sub_metr,axiom,
$true ).
fof(t25_sub_metr,axiom,
$true ).
fof(d3_sub_metr,axiom,
$true ).
fof(d4_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ( v3_sub_metr(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_xreal_0(k2_metric_1(A,B,D),k4_square_1(k2_metric_1(A,B,C),k2_metric_1(A,C,D))) ) ) ) ) ) ).
fof(t26_sub_metr,axiom,
$true ).
fof(t27_sub_metr,axiom,
$true ).
fof(t28_sub_metr,axiom,
$true ).
fof(t29_sub_metr,axiom,
$true ).
fof(t30_sub_metr,axiom,
$true ).
fof(t31_sub_metr,axiom,
$true ).
fof(t32_sub_metr,axiom,
$true ).
fof(t33_sub_metr,axiom,
$true ).
fof(t34_sub_metr,axiom,
$true ).
fof(t35_sub_metr,axiom,
$true ).
fof(t36_sub_metr,axiom,
$true ).
fof(t37_sub_metr,axiom,
$true ).
fof(t38_sub_metr,axiom,
$true ).
fof(d5_sub_metr,axiom,
k1_sub_metr = k2_funcop_1(k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))),np__0) ).
fof(t39_sub_metr,axiom,
( r2_hidden(k4_tarski(k1_xboole_0,k1_xboole_0),k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))))
& r2_hidden(k4_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))))
& r2_hidden(k4_tarski(k1_tarski(k1_xboole_0),k1_xboole_0),k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))))
& r2_hidden(k4_tarski(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0)),k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))) ) ).
fof(t40_sub_metr,axiom,
! [A] :
( m1_subset_1(A,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
=> ! [B] :
( m1_subset_1(B,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
=> k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,A,B) = np__0 ) ) ).
fof(t41_sub_metr,axiom,
$true ).
fof(t42_sub_metr,axiom,
! [A] :
( m1_subset_1(A,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
=> ! [B] :
( m1_subset_1(B,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
=> k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,A,B) = k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,B,A) ) ) ).
fof(t43_sub_metr,axiom,
! [A] :
( m1_subset_1(A,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
=> ! [B] :
( m1_subset_1(B,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
=> ! [C] :
( m1_subset_1(C,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
=> r1_xreal_0(k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,A,C),k2_xcmplx_0(k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,A,B),k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,B,C))) ) ) ) ).
fof(d6_sub_metr,axiom,
k2_sub_metr = g1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr) ).
fof(d7_sub_metr,axiom,
! [A] :
( l1_metric_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_sub_metr(A,B,C,D)
<=> ( B != C
& B != D
& C != D
& k2_metric_1(A,B,D) = k2_xcmplx_0(k2_metric_1(A,B,C),k2_metric_1(A,C,D)) ) ) ) ) ) ) ).
fof(t44_sub_metr,axiom,
$true ).
fof(t45_sub_metr,axiom,
$true ).
fof(t46_sub_metr,axiom,
$true ).
fof(t47_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_sub_metr(A,B,C,D)
=> r1_sub_metr(A,D,C,B) ) ) ) ) ) ).
fof(t48_sub_metr,axiom,
! [A] :
( ( v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_sub_metr(A,B,C,D)
=> ( ~ r1_sub_metr(A,C,B,D)
& ~ r1_sub_metr(A,B,D,C) ) ) ) ) ) ) ).
fof(t49_sub_metr,axiom,
! [A] :
( ( v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r1_sub_metr(A,B,C,D)
& r1_sub_metr(A,B,D,E) )
=> ( r1_sub_metr(A,B,C,E)
& r1_sub_metr(A,C,D,E) ) ) ) ) ) ) ) ).
fof(t50_sub_metr,axiom,
$true ).
fof(t51_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,k3_sub_metr(A,B,C))
<=> r1_sub_metr(A,B,D,C) ) ) ) ) ) ).
fof(t52_sub_metr,axiom,
$true ).
fof(t53_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,k4_sub_metr(A,B,C))
<=> ~ ( ~ r1_sub_metr(A,B,D,C)
& D != B
& D != C ) ) ) ) ) ) ).
fof(t54_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_tarski(k3_sub_metr(A,B,C),k4_sub_metr(A,B,C)) ) ) ) ).
fof(dt_k1_sub_metr,axiom,
( v1_funct_1(k1_sub_metr)
& v1_funct_2(k1_sub_metr,k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))),k1_numbers)
& m2_relset_1(k1_sub_metr,k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))),k1_numbers) ) ).
fof(dt_k2_sub_metr,axiom,
l1_metric_1(k2_sub_metr) ).
fof(dt_k3_sub_metr,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k3_sub_metr(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k4_sub_metr,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k4_sub_metr(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(d8_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k3_sub_metr(A,B,C) = a_3_0_sub_metr(A,B,C) ) ) ) ).
fof(d9_sub_metr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_sub_metr(A,B,C) = k2_xboole_0(a_3_0_sub_metr(A,B,C),k2_struct_0(A,B,C)) ) ) ) ).
fof(fraenkel_a_3_0_sub_metr,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_sub_metr(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_sub_metr(B,C,E,D) ) ) ) ).
%------------------------------------------------------------------------------