SET007 Axioms: SET007+264.ax
%------------------------------------------------------------------------------
% File : SET007+264 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Metrics in Cartesian Product
% 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 : metric_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 43 ( 12 unt; 0 def)
% Number of atoms : 675 ( 36 equ)
% Maximal formula atoms : 41 ( 15 avg)
% Number of connectives : 718 ( 86 ~; 0 |; 478 &)
% ( 6 <=>; 148 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 13 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 14 con; 0-8 aty)
% Number of variables : 170 ( 167 !; 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_metric_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
& m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
=> ( ~ v3_struct_0(g1_metric_1(A,B))
& v1_metric_1(g1_metric_1(A,B)) ) ) ).
fof(d1_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k1_numbers)
& m2_relset_1(C,k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k1_numbers) )
=> ( C = k1_metric_3(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ! [H] :
( m1_subset_1(H,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [I] :
( m1_subset_1(I,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ( ( H = k4_tarski(D,F)
& I = k4_tarski(E,G) )
=> k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),C,H,I) = k3_real_1(k4_metric_1(A,D,E),k4_metric_1(B,F,G)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_metric_3,axiom,
$true ).
fof(t2_metric_3,axiom,
$true ).
fof(t3_metric_3,axiom,
$true ).
fof(t4_metric_3,axiom,
$true ).
fof(t5_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ( k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,D) = np__0
<=> C = D ) ) ) ) ) ).
fof(t6_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,D) = k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),D,C) ) ) ) ) ).
fof(t7_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> r1_xreal_0(k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,E),k3_real_1(k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,D),k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),D,E))) ) ) ) ) ) ).
fof(d2_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
=> k2_metric_3(A,B,C,D) = k1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B),C,D) ) ) ) ) ).
fof(d3_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> k3_metric_3(A,B) = g1_metric_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k1_metric_3(A,B)) ) ) ).
fof(d4_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))),k1_numbers)
& m2_relset_1(D,k2_zfmisc_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))),k1_numbers) )
=> ( D = k4_metric_3(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(B))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(C))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(C))
=> ! [K] :
( m1_subset_1(K,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> ! [L] :
( m1_subset_1(L,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> ( ( K = k3_mcart_1(E,G,I)
& L = k3_mcart_1(F,H,J) )
=> k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),D,K,L) = k3_real_1(k3_real_1(k4_metric_1(A,E,F),k4_metric_1(B,G,H)),k4_metric_1(C,I,J)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_metric_3,axiom,
$true ).
fof(t9_metric_3,axiom,
$true ).
fof(t10_metric_3,axiom,
$true ).
fof(t11_metric_3,axiom,
$true ).
fof(t12_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> ! [E] :
( m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> ( k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,E) = np__0
<=> D = E ) ) ) ) ) ) ).
fof(t13_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> ! [E] :
( m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,E) = k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),E,D) ) ) ) ) ) ).
fof(t14_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> ! [E] :
( m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> ! [F] :
( m1_subset_1(F,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> r1_xreal_0(k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,F),k3_real_1(k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,E),k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),E,F))) ) ) ) ) ) ) ).
fof(d5_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> k5_metric_3(A,B,C) = g1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C)) ) ) ) ).
fof(d6_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> ! [E] :
( m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
=> k6_metric_3(A,B,C,D,E) = k1_metric_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k4_metric_3(A,B,C),D,E) ) ) ) ) ) ).
fof(d7_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers)
& m2_relset_1(E,k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers) )
=> ( E = k7_metric_3(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(B))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(B))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(C))
=> ! [K] :
( m1_subset_1(K,u1_struct_0(C))
=> ! [L] :
( m1_subset_1(L,u1_struct_0(D))
=> ! [M] :
( m1_subset_1(M,u1_struct_0(D))
=> ! [N] :
( m1_subset_1(N,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> ! [O] :
( m1_subset_1(O,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> ( ( N = k4_mcart_1(F,H,J,L)
& O = k4_mcart_1(G,I,K,M) )
=> k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),E,N,O) = k3_real_1(k3_real_1(k4_metric_1(A,F,G),k4_metric_1(B,H,I)),k3_real_1(k4_metric_1(C,J,K),k4_metric_1(D,L,M))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_metric_3,axiom,
$true ).
fof(t16_metric_3,axiom,
$true ).
fof(t17_metric_3,axiom,
$true ).
fof(t18_metric_3,axiom,
$true ).
fof(t19_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D) )
=> ! [E] :
( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> ! [F] :
( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> ( k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F) = np__0
<=> E = F ) ) ) ) ) ) ) ).
fof(t20_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D) )
=> ! [E] :
( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> ! [F] :
( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F) = k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),F,E) ) ) ) ) ) ) ).
fof(t21_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D) )
=> ! [E] :
( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> ! [F] :
( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> ! [G] :
( m1_subset_1(G,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> r1_xreal_0(k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,G),k3_real_1(k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F),k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),F,G))) ) ) ) ) ) ) ) ).
fof(d8_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D) )
=> k8_metric_3(A,B,C,D) = g1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D)) ) ) ) ) ).
fof(d9_metric_3,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] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D) )
=> ! [E] :
( m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> ! [F] :
( m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
=> k9_metric_3(A,B,C,D,E,F) = k1_metric_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k7_metric_3(A,B,C,D),E,F) ) ) ) ) ) ) ).
fof(s1_metric_3,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3),k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3)),f3_s1_metric_3)
& m2_relset_1(A,k2_zfmisc_1(k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3),k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3)),f3_s1_metric_3)
& ! [B] :
( m1_subset_1(B,f1_s1_metric_3)
=> ! [C] :
( m1_subset_1(C,f1_s1_metric_3)
=> ! [D] :
( m1_subset_1(D,f2_s1_metric_3)
=> ! [E] :
( m1_subset_1(E,f2_s1_metric_3)
=> ! [F] :
( m1_subset_1(F,k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3))
=> ! [G] :
( m1_subset_1(G,k2_zfmisc_1(f1_s1_metric_3,f2_s1_metric_3))
=> ( ( F = k4_tarski(B,D)
& G = k4_tarski(C,E) )
=> k1_funct_1(A,k4_tarski(F,G)) = f4_s1_metric_3(B,C,D,E) ) ) ) ) ) ) ) ) ).
fof(s2_metric_3,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3),k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3)),f4_s2_metric_3)
& m2_relset_1(A,k2_zfmisc_1(k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3),k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3)),f4_s2_metric_3)
& ! [B] :
( m1_subset_1(B,f1_s2_metric_3)
=> ! [C] :
( m1_subset_1(C,f1_s2_metric_3)
=> ! [D] :
( m1_subset_1(D,f2_s2_metric_3)
=> ! [E] :
( m1_subset_1(E,f2_s2_metric_3)
=> ! [F] :
( m1_subset_1(F,f3_s2_metric_3)
=> ! [G] :
( m1_subset_1(G,f3_s2_metric_3)
=> ! [H] :
( m1_subset_1(H,k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3))
=> ! [I] :
( m1_subset_1(I,k3_zfmisc_1(f1_s2_metric_3,f2_s2_metric_3,f3_s2_metric_3))
=> ( ( H = k3_mcart_1(B,D,F)
& I = k3_mcart_1(C,E,G) )
=> k1_funct_1(A,k4_tarski(H,I)) = f5_s2_metric_3(B,C,D,E,F,G) ) ) ) ) ) ) ) ) ) ) ).
fof(s3_metric_3,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3),k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3)),f5_s3_metric_3)
& m2_relset_1(A,k2_zfmisc_1(k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3),k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3)),f5_s3_metric_3)
& ! [B] :
( m1_subset_1(B,f1_s3_metric_3)
=> ! [C] :
( m1_subset_1(C,f1_s3_metric_3)
=> ! [D] :
( m1_subset_1(D,f2_s3_metric_3)
=> ! [E] :
( m1_subset_1(E,f2_s3_metric_3)
=> ! [F] :
( m1_subset_1(F,f3_s3_metric_3)
=> ! [G] :
( m1_subset_1(G,f3_s3_metric_3)
=> ! [H] :
( m1_subset_1(H,f4_s3_metric_3)
=> ! [I] :
( m1_subset_1(I,f4_s3_metric_3)
=> ! [J] :
( m1_subset_1(J,k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3))
=> ! [K] :
( m1_subset_1(K,k4_zfmisc_1(f1_s3_metric_3,f2_s3_metric_3,f3_s3_metric_3,f4_s3_metric_3))
=> ( ( J = k4_mcart_1(B,D,F,H)
& K = k4_mcart_1(C,E,G,I) )
=> k1_funct_1(A,k4_tarski(J,K)) = f6_s3_metric_3(B,C,D,E,F,G,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_metric_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ( v1_funct_1(k1_metric_3(A,B))
& v1_funct_2(k1_metric_3(A,B),k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k1_numbers)
& m2_relset_1(k1_metric_3(A,B),k2_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k1_numbers) ) ) ).
fof(dt_k2_metric_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B)
& m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))
& m1_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))) )
=> m1_subset_1(k2_metric_3(A,B,C,D),k1_numbers) ) ).
fof(dt_k3_metric_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ( ~ v3_struct_0(k3_metric_3(A,B))
& v1_metric_1(k3_metric_3(A,B))
& v6_metric_1(k3_metric_3(A,B))
& v7_metric_1(k3_metric_3(A,B))
& v8_metric_1(k3_metric_3(A,B))
& v9_metric_1(k3_metric_3(A,B))
& l1_metric_1(k3_metric_3(A,B)) ) ) ).
fof(dt_k4_metric_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B)
& ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ( v1_funct_1(k4_metric_3(A,B,C))
& v1_funct_2(k4_metric_3(A,B,C),k2_zfmisc_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))),k1_numbers)
& m2_relset_1(k4_metric_3(A,B,C),k2_zfmisc_1(k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))),k1_numbers) ) ) ).
fof(dt_k5_metric_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B)
& ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C) )
=> ( ~ v3_struct_0(k5_metric_3(A,B,C))
& v1_metric_1(k5_metric_3(A,B,C))
& v6_metric_1(k5_metric_3(A,B,C))
& v7_metric_1(k5_metric_3(A,B,C))
& v8_metric_1(k5_metric_3(A,B,C))
& v9_metric_1(k5_metric_3(A,B,C))
& l1_metric_1(k5_metric_3(A,B,C)) ) ) ).
fof(dt_k6_metric_3,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B)
& ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C)
& m1_subset_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C)))
& m1_subset_1(E,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C))) )
=> m1_subset_1(k6_metric_3(A,B,C,D,E),k1_numbers) ) ).
fof(dt_k7_metric_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B)
& ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C)
& ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D) )
=> ( v1_funct_1(k7_metric_3(A,B,C,D))
& v1_funct_2(k7_metric_3(A,B,C,D),k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers)
& m2_relset_1(k7_metric_3(A,B,C,D),k2_zfmisc_1(k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)),k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))),k1_numbers) ) ) ).
fof(dt_k8_metric_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B)
& ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C)
& ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D) )
=> ( ~ v3_struct_0(k8_metric_3(A,B,C,D))
& v1_metric_1(k8_metric_3(A,B,C,D))
& v6_metric_1(k8_metric_3(A,B,C,D))
& v7_metric_1(k8_metric_3(A,B,C,D))
& v8_metric_1(k8_metric_3(A,B,C,D))
& v9_metric_1(k8_metric_3(A,B,C,D))
& l1_metric_1(k8_metric_3(A,B,C,D)) ) ) ).
fof(dt_k9_metric_3,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(A)
& v6_metric_1(A)
& v7_metric_1(A)
& v8_metric_1(A)
& v9_metric_1(A)
& l1_metric_1(A)
& ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B)
& ~ v3_struct_0(C)
& v6_metric_1(C)
& v7_metric_1(C)
& v8_metric_1(C)
& v9_metric_1(C)
& l1_metric_1(C)
& ~ v3_struct_0(D)
& v6_metric_1(D)
& v7_metric_1(D)
& v8_metric_1(D)
& v9_metric_1(D)
& l1_metric_1(D)
& m1_subset_1(E,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D)))
& m1_subset_1(F,k4_zfmisc_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),u1_struct_0(D))) )
=> m1_subset_1(k9_metric_3(A,B,C,D,E,F),k1_numbers) ) ).
%------------------------------------------------------------------------------