SET007 Axioms: SET007+795.ax


%------------------------------------------------------------------------------
% File     : SET007+795 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Sorting Operators for Finite 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    : rfinseq2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   44 (   0 unit)
%            Number of atoms       :  210 (  59 equality)
%            Maximal formula depth :   19 (   6 average)
%            Number of connectives :  178 (  12 ~  ;   5  |;  49  &)
%                                         (   5 <=>; 107 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   0 propositional; 1-3 arity)
%            Number of functors    :   28 (   5 constant; 0-5 arity)
%            Number of variables   :   78 (   0 singleton;  78 !;   0 ?)
%            Maximal term depth    :    3 (   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(d1_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k1_rfinseq2(A)
          <=> ( ( k3_finseq_1(A) = np__0
               => B = np__0 )
              & ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
               => ( r2_hidden(B,k4_finseq_1(A))
                  & ! [C] :
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( ( r2_hidden(C,k4_finseq_1(A))
                                  & D = k2_seq_1(k5_numbers,k1_numbers,A,C)
                                  & E = k2_seq_1(k5_numbers,k1_numbers,A,B) )
                               => r1_xreal_0(D,E) ) ) ) )
                  & ! [C] :
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ( ( r2_hidden(C,k4_finseq_1(A))
                          & k2_seq_1(k5_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,A,B) )
                       => r1_xreal_0(B,C) ) ) ) ) ) ) ) ) )).

fof(d2_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k2_rfinseq2(A)
          <=> ( ( k3_finseq_1(A) = np__0
               => B = np__0 )
              & ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
               => ( r2_hidden(B,k4_finseq_1(A))
                  & ! [C] :
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( ( r2_hidden(C,k4_finseq_1(A))
                                  & D = k2_seq_1(k5_numbers,k1_numbers,A,C)
                                  & E = k2_seq_1(k5_numbers,k1_numbers,A,B) )
                               => r1_xreal_0(E,D) ) ) ) )
                  & ! [C] :
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ( ( r2_hidden(C,k4_finseq_1(A))
                          & k2_seq_1(k5_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,A,B) )
                       => r1_xreal_0(B,C) ) ) ) ) ) ) ) ) )).

fof(d3_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k3_rfinseq2(A) = k2_seq_1(k5_numbers,k1_numbers,A,k1_rfinseq2(A)) ) )).

fof(d4_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k4_rfinseq2(A) = k2_seq_1(k5_numbers,k1_numbers,A,k2_rfinseq2(A)) ) )).

fof(t1_rfinseq2,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(B,k3_finseq_1(A)) )
           => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_seq_1(k5_numbers,k1_numbers,A,k1_rfinseq2(A)))
              & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k3_rfinseq2(A)) ) ) ) ) )).

fof(t2_rfinseq2,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(B,k3_finseq_1(A)) )
           => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,k2_rfinseq2(A)),k2_seq_1(k5_numbers,k1_numbers,A,B))
              & r1_xreal_0(k4_rfinseq2(A),k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ) )).

fof(t3_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( A = k12_finseq_1(k1_numbers,B)
           => ( k1_rfinseq2(A) = np__1
              & k3_rfinseq2(A) = B ) ) ) ) )).

fof(t4_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( A = k12_finseq_1(k1_numbers,B)
           => ( k2_rfinseq2(A) = np__1
              & k4_rfinseq2(A) = B ) ) ) ) )).

fof(t5_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( A = k10_finseq_1(B,C)
               => ( k3_rfinseq2(A) = k4_square_1(B,C)
                  & k1_rfinseq2(A) = k2_cqc_lang(k1_numbers,B,k4_square_1(B,C),np__1,np__2) ) ) ) ) ) )).

fof(t6_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( A = k10_finseq_1(B,C)
               => ( k4_rfinseq2(A) = k3_square_1(B,C)
                  & k2_rfinseq2(A) = k2_cqc_lang(k1_numbers,B,k3_square_1(B,C),np__1,np__2) ) ) ) ) ) )).

fof(t7_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | r1_xreal_0(k3_rfinseq2(k3_rvsum_1(A,B)),k3_real_1(k3_rfinseq2(A),k3_rfinseq2(B))) ) ) ) ) )).

fof(t8_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | r1_xreal_0(k3_real_1(k4_rfinseq2(A),k4_rfinseq2(B)),k4_rfinseq2(k3_rvsum_1(A,B))) ) ) ) ) )).

fof(t9_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
              & ~ r1_xreal_0(B,np__0)
              & ~ ( k3_rfinseq2(k9_rvsum_1(B,A)) = k4_real_1(B,k3_rfinseq2(A))
                  & k1_rfinseq2(k9_rvsum_1(B,A)) = k1_rfinseq2(A) ) ) ) ) )).

fof(t10_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
              & ~ r1_xreal_0(B,np__0)
              & ~ ( k4_rfinseq2(k9_rvsum_1(B,A)) = k4_real_1(B,k4_rfinseq2(A))
                  & k2_rfinseq2(k9_rvsum_1(B,A)) = k2_rfinseq2(A) ) ) ) ) )).

fof(t11_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => ( k3_rfinseq2(k5_rvsum_1(A)) = k1_real_1(k4_rfinseq2(A))
          & k1_rfinseq2(k5_rvsum_1(A)) = k2_rfinseq2(A) ) ) ) )).

fof(t12_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => ( k4_rfinseq2(k5_rvsum_1(A)) = k1_real_1(k3_rfinseq2(A))
          & k2_rfinseq2(k5_rvsum_1(A)) = k1_rfinseq2(A) ) ) ) )).

fof(t13_rfinseq2,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)
              | ( r1_xreal_0(k3_rfinseq2(k1_rfinseq(k1_numbers,A,B)),k3_rfinseq2(A))
                & r1_xreal_0(k4_rfinseq2(A),k4_rfinseq2(k1_rfinseq(k1_numbers,A,B))) ) ) ) ) ) )).

fof(t14_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( r1_rfinseq(A,B)
           => k3_rfinseq2(A) = k3_rfinseq2(B) ) ) ) )).

fof(t15_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( r1_rfinseq(A,B)
           => k4_rfinseq2(A) = k4_rfinseq2(B) ) ) ) )).

fof(d5_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_rfinseq(B)
            & m2_finseq_1(B,k1_numbers) )
         => ( B = k5_rfinseq2(A)
          <=> r1_rfinseq(A,B) ) ) ) )).

fof(t16_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( ( k3_finseq_1(A) = np__0
          | k3_finseq_1(A) = np__1 )
       => v1_integra2(A) ) ) )).

fof(t17_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( v1_integra2(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,B),k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ) ) )).

fof(t18_rfinseq2,axiom,(
    ! [A] :
      ( ( v1_integra2(A)
        & m2_finseq_1(A,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( v1_integra2(k16_finseq_1(k1_numbers,A,B))
            & m2_finseq_1(k16_finseq_1(k1_numbers,A,B),k1_numbers) ) ) ) )).

fof(t19_rfinseq2,axiom,(
    ! [A] :
      ( ( v1_integra2(A)
        & m2_finseq_1(A,k1_numbers) )
     => ! [B] :
          ( ( v1_integra2(B)
            & m2_finseq_1(B,k1_numbers) )
         => ( r1_rfinseq(A,B)
           => A = B ) ) ) )).

fof(d6_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra2(B)
            & m2_finseq_1(B,k1_numbers) )
         => ( B = k6_rfinseq2(A)
          <=> r1_rfinseq(A,B) ) ) ) )).

fof(t20_rfinseq2,axiom,(
    ! [A] :
      ( ( v1_rfinseq(A)
        & m2_finseq_1(A,k1_numbers) )
     => k5_rfinseq2(A) = A ) )).

fof(t21_rfinseq2,axiom,(
    ! [A] :
      ( ( v1_integra2(A)
        & m2_finseq_1(A,k1_numbers) )
     => k6_rfinseq2(A) = A ) )).

fof(t22_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k5_rfinseq2(k5_rfinseq2(A)) = k5_rfinseq2(A) ) )).

fof(t23_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k6_rfinseq2(k6_rfinseq2(A)) = k6_rfinseq2(A) ) )).

fof(t24_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( v1_rfinseq(A)
       => v1_integra2(k5_rvsum_1(A)) ) ) )).

fof(t25_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( v1_integra2(A)
       => v1_rfinseq(k5_rvsum_1(A)) ) ) )).

fof(t26_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ! [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)
                  & r1_xreal_0(np__1,k3_finseq_1(B)) )
               => k5_rvsum_1(A) = k5_relat_1(C,k5_rvsum_1(B)) ) ) ) ) )).

fof(t27_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( r1_rfinseq(A,B)
           => r1_rfinseq(k5_rvsum_1(A),k5_rvsum_1(B)) ) ) ) )).

fof(t28_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k5_rfinseq2(k5_rvsum_1(A)) = k5_rvsum_1(k6_rfinseq2(A)) ) )).

fof(t29_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k6_rfinseq2(k5_rvsum_1(A)) = k5_rvsum_1(k5_rfinseq2(A)) ) )).

fof(t30_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( k4_finseq_1(k5_rfinseq2(A)) = k4_finseq_1(A)
        & k3_finseq_1(k5_rfinseq2(A)) = k3_finseq_1(A) ) ) )).

fof(t31_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( k4_finseq_1(k6_rfinseq2(A)) = k4_finseq_1(A)
        & k3_finseq_1(k6_rfinseq2(A)) = k3_finseq_1(A) ) ) )).

fof(t32_rfinseq2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( r1_xreal_0(np__1,k3_finseq_1(A))
       => ( k1_rfinseq2(k5_rfinseq2(A)) = np__1
          & k2_rfinseq2(k6_rfinseq2(A)) = np__1
          & k2_seq_1(k5_numbers,k1_numbers,k5_rfinseq2(A),np__1) = k3_rfinseq2(A)
          & k2_seq_1(k5_numbers,k1_numbers,k6_rfinseq2(A),np__1) = k4_rfinseq2(A) ) ) ) )).

fof(dt_k1_rfinseq2,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m2_subset_1(k1_rfinseq2(A),k1_numbers,k5_numbers) ) )).

fof(dt_k2_rfinseq2,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m2_subset_1(k2_rfinseq2(A),k1_numbers,k5_numbers) ) )).

fof(dt_k3_rfinseq2,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m1_subset_1(k3_rfinseq2(A),k1_numbers) ) )).

fof(dt_k4_rfinseq2,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m1_subset_1(k4_rfinseq2(A),k1_numbers) ) )).

fof(dt_k5_rfinseq2,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => ( v1_rfinseq(k5_rfinseq2(A))
        & m2_finseq_1(k5_rfinseq2(A),k1_numbers) ) ) )).

fof(dt_k6_rfinseq2,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => ( v1_integra2(k6_rfinseq2(A))
        & m2_finseq_1(k6_rfinseq2(A),k1_numbers) ) ) )).
%------------------------------------------------------------------------------