SET007 Axioms: SET007+771.ax


%------------------------------------------------------------------------------
% File     : SET007+771 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  118 (  20 equality)
%            Maximal formula depth :   18 (  11 average)
%            Number of connectives :  111 (   3 ~  ;   1  |;  57  &)
%                                         (   1 <=>;  49 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   0 propositional; 1-3 arity)
%            Number of functors    :   27 (   7 constant; 0-4 arity)
%            Number of variables   :   41 (   0 singleton;  40 !;   1 ?)
%            Maximal term depth    :    5 (   1 average)
% 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)) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------