SET007 Axioms: SET007+98.ax


%------------------------------------------------------------------------------
% File     : SET007+98 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Binary Operations on 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    : finsop_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   33 (   1 unit)
%            Number of atoms       :  290 (  53 equality)
%            Maximal formula depth :   25 (  11 average)
%            Number of connectives :  325 (  68 ~  ;  11  |; 124  &)
%                                         (   2 <=>; 120 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   1 propositional; 0-3 arity)
%            Number of functors    :   33 (   5 constant; 0-6 arity)
%            Number of variables   :  118 (   0 singleton; 117 !;   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(d1_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( ( v1_setwiseo(C,A)
                  | r1_xreal_0(np__1,k3_finseq_1(B)) )
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ( ( ( v1_setwiseo(C,A)
                          & k3_finseq_1(B) = np__0 )
                       => ( D = k2_finsop_1(A,B,C)
                        <=> D = k3_binop_1(A,C) ) )
                      & ( ~ ( v1_setwiseo(C,A)
                            & k3_finseq_1(B) = np__0 )
                       => ( D = k2_finsop_1(A,B,C)
                        <=> ? [E] :
                              ( v1_funct_1(E)
                              & v1_funct_2(E,k5_numbers,A)
                              & m2_relset_1(E,k5_numbers,A)
                              & k8_funct_2(k5_numbers,A,E,np__1) = k1_funct_1(B,np__1)
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ~ ( np__0 != F
                                      & ~ r1_xreal_0(k3_finseq_1(B),F)
                                      & k8_funct_2(k5_numbers,A,E,k1_nat_1(F,np__1)) != k1_binop_1(C,k8_funct_2(k5_numbers,A,E,F),k1_funct_1(B,k1_nat_1(F,np__1))) ) )
                              & D = k8_funct_2(k5_numbers,A,E,k3_finseq_1(B)) ) ) ) ) ) ) ) ) ) )).

fof(t1_finsop_1,axiom,(
    $true )).

fof(t2_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ~ ( r1_xreal_0(np__1,k3_finseq_1(B))
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,A)
                        & m2_relset_1(D,k5_numbers,A) )
                     => ~ ( k8_funct_2(k5_numbers,A,D,np__1) = k1_funct_1(B,np__1)
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ ( np__0 != E
                                  & ~ r1_xreal_0(k3_finseq_1(B),E)
                                  & k8_funct_2(k5_numbers,A,D,k1_nat_1(E,np__1)) != k1_binop_1(C,k8_funct_2(k5_numbers,A,D,E),k1_funct_1(B,k1_nat_1(E,np__1))) ) )
                          & k2_finsop_1(A,B,C) = k8_funct_2(k5_numbers,A,D,k3_finseq_1(B)) ) ) ) ) ) ) )).

fof(t3_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ( r1_xreal_0(np__1,k3_finseq_1(C))
                   => ( ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,A)
                            & m2_relset_1(E,k5_numbers,A) )
                         => ~ ( k8_funct_2(k5_numbers,A,E,np__1) = k1_funct_1(C,np__1)
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ~ ( np__0 != F
                                      & ~ r1_xreal_0(k3_finseq_1(C),F)
                                      & k8_funct_2(k5_numbers,A,E,k1_nat_1(F,np__1)) != k1_binop_1(D,k8_funct_2(k5_numbers,A,E,F),k1_funct_1(C,k1_nat_1(F,np__1))) ) )
                              & B = k8_funct_2(k5_numbers,A,E,k3_finseq_1(C)) ) )
                      | B = k2_finsop_1(A,C,D) ) ) ) ) ) ) )).

fof(t4_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( ( v2_binop_1(C,A)
                  & v1_binop_1(C,A) )
               => ( ( ~ v1_setwiseo(C,A)
                    & ~ r1_xreal_0(np__1,k3_finseq_1(B)) )
                  | k2_finsop_1(A,B,C) = k7_setwiseo(k5_numbers,A,C,k5_finsop_1(B),k4_finsop_1(A,k3_finsop_1(A,k5_numbers,k3_binop_1(A,C)),B)) ) ) ) ) ) )).

fof(t5_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ( ( v1_setwiseo(D,A)
                      | r1_xreal_0(np__1,k3_finseq_1(C)) )
                   => k2_finsop_1(A,k8_finseq_1(A,C,k12_finseq_1(A,B)),D) = k2_binop_1(A,A,A,D,k2_finsop_1(A,C,D),B) ) ) ) ) ) )).

fof(t6_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ( v2_binop_1(D,A)
                   => ( ( ~ v1_setwiseo(D,A)
                        & ~ ( r1_xreal_0(np__1,k3_finseq_1(B))
                            & r1_xreal_0(np__1,k3_finseq_1(C)) ) )
                      | k2_finsop_1(A,k8_finseq_1(A,B,C),D) = k2_binop_1(A,A,A,D,k2_finsop_1(A,B,D),k2_finsop_1(A,C,D)) ) ) ) ) ) ) )).

fof(t7_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ( v2_binop_1(D,A)
                   => ( ( ~ v1_setwiseo(D,A)
                        & ~ r1_xreal_0(np__1,k3_finseq_1(C)) )
                      | k2_finsop_1(A,k8_finseq_1(A,k12_finseq_1(A,B),C),D) = k2_binop_1(A,A,A,D,B,k2_finsop_1(A,C,D)) ) ) ) ) ) ) )).

fof(t8_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B))
                        & v3_funct_2(E,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B))
                        & m2_relset_1(E,k4_relset_1(k5_numbers,A,B),k4_relset_1(k5_numbers,A,B)) )
                     => ( ( v1_binop_1(D,A)
                          & v2_binop_1(D,A)
                          & C = k5_relat_1(E,B) )
                       => ( ( ~ v1_setwiseo(D,A)
                            & ~ r1_xreal_0(np__1,k3_finseq_1(B)) )
                          | k2_finsop_1(A,B,D) = k2_finsop_1(A,C,D) ) ) ) ) ) ) ) )).

fof(t9_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ( ( v2_binop_1(D,A)
                      & v1_binop_1(D,A)
                      & v2_funct_1(B)
                      & v2_funct_1(C)
                      & k2_relat_1(B) = k2_relat_1(C) )
                   => ( ( ~ v1_setwiseo(D,A)
                        & ~ r1_xreal_0(np__1,k3_finseq_1(B)) )
                      | k2_finsop_1(A,B,D) = k2_finsop_1(A,C,D) ) ) ) ) ) ) )).

fof(t10_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( m2_finseq_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,A),A)
                        & m2_relset_1(E,k2_zfmisc_1(A,A),A) )
                     => ( ( v2_binop_1(E,A)
                          & v1_binop_1(E,A)
                          & k3_finseq_1(B) = k3_finseq_1(C)
                          & k3_finseq_1(B) = k3_finseq_1(D)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ( r2_hidden(F,k5_finsop_1(B))
                               => k1_funct_1(B,F) = k1_binop_1(E,k1_funct_1(C,F),k1_funct_1(D,F)) ) ) )
                       => ( ( ~ v1_setwiseo(E,A)
                            & ~ r1_xreal_0(np__1,k3_finseq_1(B)) )
                          | k2_finsop_1(A,B,E) = k2_binop_1(A,A,A,E,k2_finsop_1(A,C,E),k2_finsop_1(A,D,E)) ) ) ) ) ) ) ) )).

fof(t11_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( v1_setwiseo(B,A)
           => k2_finsop_1(A,k6_finseq_1(A),B) = k3_binop_1(A,B) ) ) ) )).

fof(t12_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => k2_finsop_1(A,k12_finseq_1(A,B),C) = B ) ) ) )).

fof(t13_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => k2_finsop_1(A,k2_finseq_4(A,B,C),D) = k2_binop_1(A,A,A,D,B,C) ) ) ) ) )).

fof(t14_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ( v1_binop_1(D,A)
                   => k2_finsop_1(A,k2_finseq_4(A,B,C),D) = k2_finsop_1(A,k2_finseq_4(A,C,B),D) ) ) ) ) ) )).

fof(t15_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,A),A)
                        & m2_relset_1(E,k2_zfmisc_1(A,A),A) )
                     => k2_finsop_1(A,k3_finseq_4(A,B,C,D),E) = k2_binop_1(A,A,A,E,k2_binop_1(A,A,A,E,B,C),D) ) ) ) ) ) )).

fof(t16_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,A),A)
                        & m2_relset_1(E,k2_zfmisc_1(A,A),A) )
                     => ( v1_binop_1(E,A)
                       => k2_finsop_1(A,k3_finseq_4(A,B,C,D),E) = k2_finsop_1(A,k3_finseq_4(A,C,B,D),E) ) ) ) ) ) ) )).

fof(t17_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => k2_finsop_1(A,k1_finsop_1(A,np__1,B),C) = B ) ) ) )).

fof(t18_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => k2_finsop_1(A,k1_finsop_1(A,np__2,B),C) = k2_binop_1(A,A,A,C,B,B) ) ) ) )).

fof(t19_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( v2_binop_1(C,A)
                       => ( ( ~ v1_setwiseo(C,A)
                            & ~ ( D != np__0
                                & E != np__0 ) )
                          | k2_finsop_1(A,k1_finsop_1(A,k1_nat_1(D,E),B),C) = k2_binop_1(A,A,A,C,k2_finsop_1(A,k1_finsop_1(A,D,B),C),k2_finsop_1(A,k1_finsop_1(A,E,B),C)) ) ) ) ) ) ) ) )).

fof(t20_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( v2_binop_1(C,A)
                       => ( ( ~ v1_setwiseo(C,A)
                            & ~ ( D != np__0
                                & E != np__0 ) )
                          | k2_finsop_1(A,k1_finsop_1(A,k2_nat_1(D,E),B),C) = k2_finsop_1(A,k1_finsop_1(A,E,k2_finsop_1(A,k1_finsop_1(A,D,B),C)),C) ) ) ) ) ) ) ) )).

fof(t21_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( k3_finseq_1(B) = np__1
               => k2_finsop_1(A,B,C) = k1_funct_1(B,np__1) ) ) ) ) )).

fof(t22_finsop_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( k3_finseq_1(B) = np__2
               => k2_finsop_1(A,B,C) = k1_binop_1(C,k1_funct_1(B,np__1),k1_funct_1(B,np__2)) ) ) ) ) )).

fof(dt_k1_finsop_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,A) )
     => m2_finseq_1(k1_finsop_1(A,B,C),A) ) )).

fof(redefinition_k1_finsop_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,A) )
     => k1_finsop_1(A,B,C) = k2_finseq_2(B,C) ) )).

fof(dt_k2_finsop_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A) )
     => m1_subset_1(k2_finsop_1(A,B,C),A) ) )).

fof(dt_k3_finsop_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,A) )
     => ( v1_funct_1(k3_finsop_1(A,B,C))
        & v1_funct_2(k3_finsop_1(A,B,C),B,A)
        & m2_relset_1(k3_finsop_1(A,B,C),B,A) ) ) )).

fof(redefinition_k3_finsop_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,A) )
     => k3_finsop_1(A,B,C) = k2_funcop_1(B,C) ) )).

fof(dt_k4_finsop_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,A)
        & m1_relset_1(B,k5_numbers,A)
        & m1_finseq_1(C,A) )
     => ( v1_funct_1(k4_finsop_1(A,B,C))
        & v1_funct_2(k4_finsop_1(A,B,C),k5_numbers,A)
        & m2_relset_1(k4_finsop_1(A,B,C),k5_numbers,A) ) ) )).

fof(idempotence_k4_finsop_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,A)
        & m1_relset_1(B,k5_numbers,A)
        & m1_finseq_1(C,A) )
     => k4_finsop_1(A,B,B) = B ) )).

fof(redefinition_k4_finsop_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,A)
        & m1_relset_1(B,k5_numbers,A)
        & m1_finseq_1(C,A) )
     => k4_finsop_1(A,B,C) = k1_funct_4(B,C) ) )).

fof(dt_k5_finsop_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => m1_subset_1(k5_finsop_1(A),k5_finsub_1(k5_numbers)) ) )).

fof(redefinition_k5_finsop_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => k5_finsop_1(A) = k1_relat_1(A) ) )).
%------------------------------------------------------------------------------