SET007 Axioms: SET007+165.ax


%------------------------------------------------------------------------------
% File     : SET007+165 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Functions and Finite Sequences of Real Numbers
% 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    : rfinseq [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   48 (   3 unit)
%            Number of atoms       :  290 (  52 equality)
%            Maximal formula depth :   15 (   7 average)
%            Number of connectives :  250 (   8 ~  ;   3  |; 103  &)
%                                         (  15 <=>; 121 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   1 propositional; 0-3 arity)
%            Number of functors    :   33 (   5 constant; 0-4 arity)
%            Number of variables   :  113 (   0 singleton; 107 !;   6 ?)
%            Maximal term depth    :    4 (   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_rfinseq,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(B) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B))
        & v1_finset_1(k7_relat_1(A,B)) ) ) )).

fof(rc1_rfinseq,axiom,(
    ? [A] :
      ( m1_finseq_1(A,k1_numbers)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A)
      & v1_rfinseq(A) ) )).

fof(d1_rfinseq,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( r1_rfinseq(A,B)
          <=> ! [C] : k1_card_1(k10_relat_1(A,k1_tarski(C))) = k1_card_1(k10_relat_1(B,k1_tarski(C))) ) ) ) )).

fof(t1_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_rfinseq(A,B)
           => k2_relat_1(A) = k2_relat_1(B) ) ) ) )).

fof(t2_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( r1_rfinseq(A,B)
                  & r1_rfinseq(A,C) )
               => r1_rfinseq(B,C) ) ) ) ) )).

fof(t3_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_rfinseq(A,B)
          <=> ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & k1_relat_1(C) = k1_relat_1(A)
                & k2_relat_1(C) = k1_relat_1(B)
                & v2_funct_1(C)
                & A = k5_relat_1(C,B) ) ) ) ) )).

fof(t4_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_rfinseq(A,B)
          <=> ! [C] : k1_card_1(k10_relat_1(A,C)) = k1_card_1(k10_relat_1(B,C)) ) ) ) )).

fof(t5_rfinseq,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( r1_tarski(k2_relat_1(B),A)
                  & r1_tarski(k2_relat_1(C),A) )
               => ( r1_rfinseq(B,C)
                <=> ! [D] :
                      ( m1_subset_1(D,A)
                     => k1_card_1(k10_relat_1(B,k1_tarski(D))) = k1_card_1(k10_relat_1(C,k1_tarski(D))) ) ) ) ) ) ) )).

fof(t6_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( k1_relat_1(A) = k1_relat_1(B)
           => ( r1_rfinseq(A,B)
            <=> ? [C] :
                  ( v1_funct_1(C)
                  & v1_funct_2(C,k1_relat_1(A),k1_relat_1(A))
                  & v3_funct_2(C,k1_relat_1(A),k1_relat_1(A))
                  & m2_relset_1(C,k1_relat_1(A),k1_relat_1(A))
                  & A = k5_relat_1(C,B) ) ) ) ) ) )).

fof(t7_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_rfinseq(A,B)
           => k1_card_1(k1_relat_1(A)) = k1_card_1(k1_relat_1(B)) ) ) ) )).

fof(t8_rfinseq,axiom,(
    $true )).

fof(t9_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finset_1(B) )
         => ( r1_rfinseq(A,B)
          <=> ! [C] : k4_card_1(k10_relat_1(A,C)) = k4_card_1(k10_relat_1(B,C)) ) ) ) )).

fof(t10_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finset_1(B) )
         => ( r1_rfinseq(A,B)
           => k4_card_1(k1_relat_1(A)) = k4_card_1(k1_relat_1(B)) ) ) ) )).

fof(t11_rfinseq,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finset_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finset_1(C) )
             => ( ( r1_tarski(k2_relat_1(B),A)
                  & r1_tarski(k2_relat_1(C),A) )
               => ( r1_rfinseq(B,C)
                <=> ! [D] :
                      ( m1_subset_1(D,A)
                     => k4_card_1(k10_relat_1(B,k1_tarski(D))) = k4_card_1(k10_relat_1(C,k1_tarski(D))) ) ) ) ) ) ) )).

fof(t12_rfinseq,axiom,(
    $true )).

fof(t13_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( r1_rfinseq(A,B)
          <=> ! [C] : k4_card_1(k10_relat_1(A,C)) = k4_card_1(k10_relat_1(B,C)) ) ) ) )).

fof(t14_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( r1_rfinseq(A,B)
              <=> r1_rfinseq(k7_finseq_1(A,C),k7_finseq_1(B,C)) ) ) ) ) )).

fof(t15_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => r1_rfinseq(k7_finseq_1(A,B),k7_finseq_1(B,A)) ) ) )).

fof(t16_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( r1_rfinseq(A,B)
           => ( k3_finseq_1(A) = k3_finseq_1(B)
              & k4_finseq_1(A) = k4_finseq_1(B) ) ) ) ) )).

fof(t17_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( r1_rfinseq(A,B)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,k4_finseq_1(B),k4_finseq_1(B))
                & v3_funct_2(C,k4_finseq_1(B),k4_finseq_1(B))
                & m2_relset_1(C,k4_finseq_1(B),k4_finseq_1(B))
                & A = k5_relat_1(C,B) ) ) ) ) )).

fof(t18_rfinseq,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( v1_finset_1(B)
         => ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & v1_finseq_1(C)
              & r1_rfinseq(k7_relat_1(A,B),C) ) ) ) )).

fof(d2_rfinseq,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ! [D] :
              ( m2_finseq_1(D,A)
             => ( ( r1_xreal_0(C,k3_finseq_1(B))
                 => ( D = k1_rfinseq(A,B,C)
                  <=> ( k3_finseq_1(D) = k5_real_1(k3_finseq_1(B),C)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k4_finseq_1(D))
                           => k1_funct_1(D,E) = k1_funct_1(B,k1_nat_1(E,C)) ) ) ) ) )
                & ( ~ r1_xreal_0(C,k3_finseq_1(B))
                 => ( D = k1_rfinseq(A,B,C)
                  <=> D = k6_finseq_1(A) ) ) ) ) ) ) )).

fof(t19_rfinseq,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r2_hidden(C,k4_finseq_1(B))
                      & r2_hidden(D,k2_finseq_1(C)) )
                   => ( k1_funct_1(k16_finseq_1(A,B,C),D) = k1_funct_1(B,D)
                      & r2_hidden(D,k4_finseq_1(B)) ) ) ) ) ) ) )).

fof(t20_rfinseq,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( k3_finseq_1(B) = k1_nat_1(C,np__1)
                    & D = k1_funct_1(B,k1_nat_1(C,np__1)) )
                 => B = k7_finseq_1(k16_finseq_1(A,B,C),k9_finseq_1(D)) ) ) ) ) )).

fof(t21_rfinseq,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k8_finseq_1(A,k16_finseq_1(A,B,C),k1_rfinseq(A,B,C)) = B ) ) ) )).

fof(t22_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( r1_rfinseq(A,B)
           => k15_rvsum_1(A) = k15_rvsum_1(B) ) ) ) )).

fof(d3_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( B = k2_rfinseq(A)
          <=> ( k3_finseq_1(B) = k3_finseq_1(A)
              & k2_seq_1(k5_numbers,k1_numbers,B,k3_finseq_1(B)) = k2_seq_1(k5_numbers,k1_numbers,A,k3_finseq_1(A))
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(np__1,C)
                      & r1_xreal_0(C,k5_real_1(k3_finseq_1(B),np__1)) )
                   => k2_seq_1(k5_numbers,k1_numbers,B,C) = k5_real_1(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) ) ) ) ) ) )).

fof(t23_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( k3_finseq_1(A) = k1_nat_1(C,np__2)
                  & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1)) = B )
               => k2_rfinseq(k16_finseq_1(k1_numbers,A,k1_nat_1(C,np__1))) = k8_finseq_1(k1_numbers,k16_finseq_1(k1_numbers,k2_rfinseq(A),C),k12_finseq_1(k1_numbers,B)) ) ) ) ) )).

fof(t24_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( k3_finseq_1(A) = k1_nat_1(D,np__2)
                      & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(D,np__1)) = B
                      & k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(D,np__2)) = C )
                   => k2_rfinseq(A) = k8_finseq_1(k1_numbers,k16_finseq_1(k1_numbers,k2_rfinseq(A),D),k2_finseq_4(k1_numbers,k5_real_1(B,C),C)) ) ) ) ) ) )).

fof(t25_rfinseq,axiom,(
    k2_rfinseq(k6_finseq_1(k1_numbers)) = k6_finseq_1(k1_numbers) )).

fof(t26_rfinseq,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k2_rfinseq(k12_finseq_1(k1_numbers,A)) = k12_finseq_1(k1_numbers,A) ) )).

fof(t27_rfinseq,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k2_rfinseq(k2_finseq_4(k1_numbers,A,B)) = k2_finseq_4(k1_numbers,k5_real_1(A,B),B) ) ) )).

fof(t28_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_rfinseq(k1_numbers,k2_rfinseq(A),B) = k2_rfinseq(k1_rfinseq(k1_numbers,A,B)) ) ) )).

fof(t29_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( k3_finseq_1(A) != np__0
       => k15_rvsum_1(k2_rfinseq(A)) = k2_seq_1(k5_numbers,k1_numbers,A,np__1) ) ) )).

fof(t30_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,B)
           => ( r1_xreal_0(k3_finseq_1(A),B)
              | k15_rvsum_1(k2_rfinseq(k1_rfinseq(k1_numbers,A,B))) = k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,np__1)) ) ) ) ) )).

fof(d4_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( v1_rfinseq(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( ( r2_hidden(B,k4_finseq_1(A))
                & r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A)) )
             => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,np__1)),k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ) )).

fof(t31_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( ( k3_finseq_1(A) = np__0
          | k3_finseq_1(A) = np__1 )
       => v1_rfinseq(A) ) ) )).

fof(t32_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( v1_rfinseq(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(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ) ) ) )).

fof(t33_rfinseq,axiom,(
    ! [A] :
      ( ( v1_rfinseq(A)
        & m2_finseq_1(A,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( v1_rfinseq(k16_finseq_1(k1_numbers,A,B))
            & m2_finseq_1(k16_finseq_1(k1_numbers,A,B),k1_numbers) ) ) ) )).

fof(t34_rfinseq,axiom,(
    ! [A] :
      ( ( v1_rfinseq(A)
        & m2_finseq_1(A,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( v1_rfinseq(k1_rfinseq(k1_numbers,A,B))
            & m2_finseq_1(k1_rfinseq(k1_numbers,A,B),k1_numbers) ) ) ) )).

fof(t35_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ? [B] :
          ( v1_rfinseq(B)
          & m2_finseq_1(B,k1_numbers)
          & r1_rfinseq(A,B) ) ) )).

fof(t36_rfinseq,axiom,(
    ! [A] :
      ( ( v1_rfinseq(A)
        & m2_finseq_1(A,k1_numbers) )
     => ! [B] :
          ( ( v1_rfinseq(B)
            & m2_finseq_1(B,k1_numbers) )
         => ( r1_rfinseq(A,B)
           => A = B ) ) ) )).

fof(t37_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( B != np__0
               => k3_funct_2(k5_numbers,k1_numbers,A,k1_tarski(k6_real_1(C,B))) = k3_funct_2(k5_numbers,k1_numbers,k9_rvsum_1(B,A),k1_tarski(C)) ) ) ) ) )).

fof(t38_rfinseq,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k3_funct_2(k5_numbers,k1_numbers,k9_rvsum_1(np__0,A),k1_tarski(np__0)) = k4_finseq_1(A) ) )).

fof(symmetry_r1_rfinseq,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => ( r1_rfinseq(A,B)
       => r1_rfinseq(B,A) ) ) )).

fof(reflexivity_r1_rfinseq,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => r1_rfinseq(A,A) ) )).

fof(dt_k1_rfinseq,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_1(k1_rfinseq(A,B,C),A) ) )).

fof(dt_k2_rfinseq,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m2_finseq_1(k2_rfinseq(A),k1_numbers) ) )).
%------------------------------------------------------------------------------