SET007 Axioms: SET007+531.ax
%------------------------------------------------------------------------------
% File : SET007+531 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Lebesgue's Covering Lemma
% Version : [Urb08] axioms.
% English : Uniform Continuity and Segmentation of Arcs
% 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 : uniform1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 18 ( 2 unt; 0 def)
% Number of atoms : 261 ( 24 equ)
% Maximal formula atoms : 28 ( 14 avg)
% Number of connectives : 298 ( 55 ~; 0 |; 148 &)
% ( 2 <=>; 93 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 16 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 30 ( 28 usr; 1 prp; 0-3 aty)
% Number of functors : 31 ( 31 usr; 6 con; 0-4 aty)
% Number of variables : 85 ( 84 !; 1 ?)
% 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(t1_uniform1,axiom,
$true ).
fof(t2_uniform1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(B,np__0)
& ~ r1_xreal_0(A,k6_real_1(np__1,B)) ) ) ) ) ).
fof(d1_uniform1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_metric_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_metric_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_uniform1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ~ ( ~ r1_xreal_0(E,k2_metric_1(A,F,G))
& r1_xreal_0(D,k2_metric_1(B,k4_finseq_4(u1_struct_0(A),u1_struct_0(B),C,F),k4_finseq_4(u1_struct_0(A),u1_struct_0(B),C,G))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t3_uniform1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(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,u1_struct_0(A),u1_struct_0(k5_pcomps_1(B)))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(k5_pcomps_1(B))) )
=> ( v5_pre_topc(C,A,k5_pcomps_1(B))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B))))
=> ( F = k9_metric_1(B,E,D)
=> v3_pre_topc(k5_pre_topc(A,k5_pcomps_1(B),C,F),A) ) ) ) ) ) ) ) ) ).
fof(t4_uniform1,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,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
& m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
=> ( ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ~ ( ~ r1_xreal_0(D,np__0)
& F = k1_funct_1(C,E)
& ! [G] :
( v1_xreal_0(G)
=> ~ ( ~ r1_xreal_0(G,np__0)
& ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(B))
=> ~ ( I = k1_funct_1(C,H)
& ~ r1_xreal_0(G,k4_metric_1(A,E,H))
& r1_xreal_0(D,k4_metric_1(B,F,I)) ) ) ) ) ) ) ) ) )
=> v5_pre_topc(C,k5_pcomps_1(A),k5_pcomps_1(B)) ) ) ) ) ).
fof(t5_uniform1,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,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
& m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
=> ( v5_pre_topc(C,k5_pcomps_1(A),k5_pcomps_1(B))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ~ ( ~ r1_xreal_0(D,np__0)
& F = k1_funct_1(C,E)
& ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( ~ r1_xreal_0(G,np__0)
& ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(B))
=> ~ ( I = k1_funct_1(C,H)
& ~ r1_xreal_0(G,k4_metric_1(A,E,H))
& r1_xreal_0(D,k4_metric_1(B,F,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t6_uniform1,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,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
& m2_relset_1(D,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
=> ( ( C = D
& v1_uniform1(C,A,B) )
=> v5_pre_topc(D,k5_pcomps_1(A),k5_pcomps_1(B)) ) ) ) ) ) ).
fof(t7_uniform1,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,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))))
=> ~ ( r1_pre_topc(k5_pcomps_1(A),B)
& v1_tops_2(B,k5_pcomps_1(A))
& v2_compts_1(k5_pcomps_1(A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( ~ r1_xreal_0(C,k4_metric_1(A,D,E))
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
=> ~ ( r2_hidden(D,F)
& r2_hidden(E,F)
& r2_hidden(F,B) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_uniform1,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,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
& m2_relset_1(D,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
=> ( ( D = C
& v2_compts_1(k5_pcomps_1(A))
& v5_pre_topc(D,k5_pcomps_1(A),k5_pcomps_1(B)) )
=> v1_uniform1(C,A,B) ) ) ) ) ) ).
fof(t9_uniform1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k14_euclid(A)))
& m2_relset_1(C,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k14_euclid(A))) )
=> ( ( v5_pre_topc(B,k5_topmetr,k15_euclid(A))
& C = B )
=> v1_uniform1(C,k2_topmetr(np__0,np__1),k14_euclid(A)) ) ) ) ) ).
fof(t10_uniform1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B)))
& m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k1_topmetr(k14_euclid(A),C)))
& m2_relset_1(E,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k1_topmetr(k14_euclid(A),C))) )
=> ( ( B = C
& v5_pre_topc(D,k5_topmetr,k3_pre_topc(k15_euclid(A),B))
& E = D )
=> v1_uniform1(E,k2_topmetr(np__0,np__1),k1_topmetr(k14_euclid(A),C)) ) ) ) ) ) ) ).
fof(t11_uniform1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
& m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k14_euclid(A)))
& m2_relset_1(C,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k14_euclid(A)))
& C = B ) ) ) ).
fof(t12_uniform1,axiom,
$true ).
fof(t13_uniform1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k18_complex1(k5_real_1(A,B)) = k18_complex1(k5_real_1(B,A)) ) ) ).
fof(t14_uniform1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r2_hidden(A,k1_rcomp_1(C,D))
& r2_hidden(B,k1_rcomp_1(C,D)) )
=> r1_xreal_0(k18_complex1(k5_real_1(A,B)),k5_real_1(D,C)) ) ) ) ) ) ).
fof(d2_uniform1,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( v2_uniform1(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(B,k4_finseq_1(A))
& r2_hidden(C,k4_finseq_1(A))
& ~ r1_xreal_0(C,B)
& r1_xreal_0(k1_goboard1(A,B),k1_goboard1(A,C)) ) ) ) ) ) ).
fof(t15_uniform1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ~ ( ~ r1_xreal_0(B,np__0)
& v5_pre_topc(C,k5_topmetr,k15_euclid(A))
& v2_funct_1(C)
& k1_funct_1(C,np__0) = D
& k1_funct_1(C,np__1) = E
& ! [F] :
( m2_finseq_1(F,k1_numbers)
=> ~ ( k1_goboard1(F,np__1) = np__0
& k1_goboard1(F,k3_finseq_1(F)) = np__1
& r1_xreal_0(np__5,k3_finseq_1(F))
& r1_tarski(k2_relat_1(F),u1_struct_0(k5_topmetr))
& v1_goboard1(F)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
=> ! [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
=> ~ ( r1_xreal_0(np__1,G)
& ~ r1_xreal_0(k3_finseq_1(F),G)
& H = k1_rcomp_1(k4_finseq_4(k5_numbers,k1_numbers,F,G),k4_finseq_4(k5_numbers,k1_numbers,F,k1_nat_1(G,np__1)))
& I = k4_pre_topc(k5_topmetr,k15_euclid(A),C,H)
& r1_xreal_0(B,k2_tbsp_1(k14_euclid(A),I)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t16_uniform1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
& m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ~ ( ~ r1_xreal_0(B,np__0)
& v5_pre_topc(C,k5_topmetr,k15_euclid(A))
& v2_funct_1(C)
& k1_funct_1(C,np__0) = D
& k1_funct_1(C,np__1) = E
& ! [F] :
( m2_finseq_1(F,k1_numbers)
=> ~ ( k1_goboard1(F,np__1) = np__1
& k1_goboard1(F,k3_finseq_1(F)) = np__0
& r1_xreal_0(np__5,k3_finseq_1(F))
& r1_tarski(k2_relat_1(F),u1_struct_0(k5_topmetr))
& v2_uniform1(F)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
=> ! [I] :
( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
=> ~ ( r1_xreal_0(np__1,G)
& ~ r1_xreal_0(k3_finseq_1(F),G)
& H = k1_rcomp_1(k4_finseq_4(k5_numbers,k1_numbers,F,k1_nat_1(G,np__1)),k4_finseq_4(k5_numbers,k1_numbers,F,G))
& I = k4_pre_topc(k5_topmetr,k15_euclid(A),C,H)
& r1_xreal_0(B,k2_tbsp_1(k14_euclid(A),I)) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------