SET007 Axioms: SET007+771.ax
%------------------------------------------------------------------------------
% File : SET007+771 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Hilbert Space of Real Sequences
% 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 : rsspace2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 10 ( 0 unt; 0 def)
% Number of atoms : 118 ( 20 equ)
% Maximal formula atoms : 27 ( 11 avg)
% Number of connectives : 111 ( 3 ~; 1 |; 57 &)
% ( 1 <=>; 49 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 11 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 20 ( 19 usr; 0 prp; 1-3 aty)
% Number of functors : 27 ( 27 usr; 7 con; 0-4 aty)
% Number of variables : 41 ( 40 !; 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(fc1_rsspace2,axiom,
( ~ v3_struct_0(k13_rsspace)
& v3_rlvect_1(k13_rsspace)
& v4_rlvect_1(k13_rsspace)
& v5_rlvect_1(k13_rsspace)
& v6_rlvect_1(k13_rsspace)
& v7_rlvect_1(k13_rsspace)
& v2_bhsp_1(k13_rsspace) ) ).
fof(fc2_rsspace2,axiom,
( ~ v3_struct_0(k13_rsspace)
& v3_rlvect_1(k13_rsspace)
& v4_rlvect_1(k13_rsspace)
& v5_rlvect_1(k13_rsspace)
& v6_rlvect_1(k13_rsspace)
& v7_rlvect_1(k13_rsspace)
& v2_bhsp_1(k13_rsspace)
& v3_bhsp_3(k13_rsspace)
& v4_bhsp_3(k13_rsspace) ) ).
fof(t1_rsspace2,axiom,
( u1_struct_0(k13_rsspace) = k11_rsspace
& ! [A] :
( m1_subset_1(A,u1_struct_0(k13_rsspace))
<=> ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& v1_series_1(k11_seq_1(k2_rsspace(A),k2_rsspace(A))) ) )
& k1_rlvect_1(k13_rsspace) = k6_rsspace
& ! [A] :
( m1_subset_1(A,u1_struct_0(k13_rsspace))
=> A = k2_rsspace(A) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k13_rsspace))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k13_rsspace))
=> k4_rlvect_1(k13_rsspace,A,B) = k9_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
& ! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k13_rsspace))
=> k3_rlvect_1(k13_rsspace,B,A) = k14_seq_1(k2_rsspace(B),A) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k13_rsspace))
=> ( k5_rlvect_1(k13_rsspace,A) = k17_seq_1(k2_rsspace(A))
& k2_rsspace(k5_rlvect_1(k13_rsspace,A)) = k17_seq_1(k2_rsspace(A)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k13_rsspace))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k13_rsspace))
=> k6_rlvect_1(k13_rsspace,A,B) = k10_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
& ! [A] :
( m1_subset_1(A,u1_struct_0(k13_rsspace))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k13_rsspace))
=> ( v1_series_1(k11_seq_1(k2_rsspace(A),k2_rsspace(B)))
& ! [C] :
( m1_subset_1(C,u1_struct_0(k13_rsspace))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k13_rsspace))
=> k1_bhsp_1(k13_rsspace,C,D) = k2_series_1(k11_seq_1(k2_rsspace(C),k2_rsspace(D))) ) ) ) ) ) ) ).
fof(t2_rsspace2,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k13_rsspace))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k13_rsspace))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k13_rsspace))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( k1_bhsp_1(k13_rsspace,A,A) = np__0
=> A = k1_rlvect_1(k13_rsspace) )
& ( A = k1_rlvect_1(k13_rsspace)
=> k1_bhsp_1(k13_rsspace,A,A) = np__0 )
& r1_xreal_0(np__0,k1_bhsp_1(k13_rsspace,A,A))
& k1_bhsp_1(k13_rsspace,A,B) = k1_bhsp_1(k13_rsspace,B,A)
& k1_bhsp_1(k13_rsspace,k4_rlvect_1(k13_rsspace,A,B),C) = k3_real_1(k1_bhsp_1(k13_rsspace,A,C),k1_bhsp_1(k13_rsspace,B,C))
& k1_bhsp_1(k13_rsspace,k3_rlvect_1(k13_rsspace,A,D),B) = k4_real_1(D,k1_bhsp_1(k13_rsspace,A,B)) ) ) ) ) ) ).
fof(t3_rsspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k13_rsspace))
& m2_relset_1(A,k5_numbers,u1_struct_0(k13_rsspace)) )
=> ( v1_bhsp_3(A,k13_rsspace)
=> v1_bhsp_2(A,k13_rsspace) ) ) ).
fof(t4_rsspace2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,B)) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B)) )
& ( v1_series_1(A)
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B),k2_series_1(A)) )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_series_1(A)) ) ) ) ) ) ) ).
fof(t5_rsspace2,axiom,
( ! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> r1_xreal_0(k4_real_1(k3_real_1(A,B),k3_real_1(A,B)),k3_real_1(k4_real_1(k4_real_1(np__2,A),A),k4_real_1(k4_real_1(np__2,B),B))) ) )
& ! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> r1_xreal_0(k4_real_1(A,A),k3_real_1(k4_real_1(k4_real_1(np__2,k5_real_1(A,B)),k5_real_1(A,B)),k4_real_1(k4_real_1(np__2,B),B))) ) ) ) ).
fof(t6_rsspace2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(B)
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_xreal_0(C,D)
& ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) )
| r1_xreal_0(k2_seq_2(B),A) ) ) ) ) ).
fof(t7_rsspace2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_real_1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),A),k5_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),A)) )
=> ( v4_seq_2(C)
& k2_seq_2(C) = k4_real_1(k5_real_1(k2_seq_2(B),A),k5_real_1(k2_seq_2(B),A)) ) ) ) ) ) ) ).
fof(t8_rsspace2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(B)
& v4_seq_2(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k3_real_1(k4_real_1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,B,E),A),k5_real_1(k2_seq_1(k5_numbers,k1_numbers,B,E),A)),k2_seq_1(k5_numbers,k1_numbers,C,E)) )
=> ( v4_seq_2(D)
& k2_seq_2(D) = k3_real_1(k4_real_1(k5_real_1(k2_seq_2(B),A),k5_real_1(k2_seq_2(B),A)),k2_seq_2(C)) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------