SET007 Axioms: SET007+386.ax


%------------------------------------------------------------------------------
% File     : SET007+386 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Free Universal Algebra Construction
% 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    : freealg [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   70 (   1 unit)
%            Number of atoms       :  479 (  38 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  538 ( 129 ~  ;   0  |; 274  &)
%                                         (  14 <=>; 121 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   42 (   1 propositional; 0-3 arity)
%            Number of functors    :   54 (   4 constant; 0-5 arity)
%            Number of variables   :  170 (   0 singleton; 158 !;  12 ?)
%            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(rc1_freealg,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_freealg(A) ) )).

fof(rc2_freealg,axiom,(
    ? [A] :
      ( m1_finseq_1(A,k5_numbers)
      & ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & ~ v2_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) )).

fof(rc3_freealg,axiom,(
    ? [A] :
      ( m1_finseq_1(A,k5_numbers)
      & ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v2_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) )).

fof(rc4_freealg,axiom,(
    ? [A] :
      ( l1_unialg_1(A)
      & ~ v3_struct_0(A)
      & v3_unialg_1(A)
      & v6_unialg_1(A)
      & v7_unialg_1(A)
      & v8_unialg_1(A)
      & v3_freealg(A) ) )).

fof(rc5_freealg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & v3_freealg(A)
        & l1_unialg_1(A) )
     => ? [B] :
          ( m1_freealg(B,A)
          & v2_freealg(B,A) ) ) )).

fof(fc1_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers) )
     => ( ~ v3_struct_0(k3_freealg(A,B))
        & v1_lang1(k3_freealg(A,B)) ) ) )).

fof(fc2_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers) )
     => ( ~ v3_struct_0(k3_freealg(A,B))
        & v1_lang1(k3_freealg(A,B))
        & v2_dtconstr(k3_freealg(A,B)) ) ) )).

fof(fc3_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers) )
     => ( ~ v3_struct_0(k3_freealg(A,B))
        & v1_lang1(k3_freealg(A,B))
        & v2_dtconstr(k3_freealg(A,B))
        & v3_dtconstr(k3_freealg(A,B)) ) ) )).

fof(fc4_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => ( ~ v3_struct_0(k3_freealg(A,B))
        & v1_lang1(k3_freealg(A,B))
        & v1_dtconstr(k3_freealg(A,B))
        & v2_dtconstr(k3_freealg(A,B))
        & v3_dtconstr(k3_freealg(A,B)) ) ) )).

fof(fc5_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => ( ~ v3_struct_0(k7_freealg(A,B))
        & v3_unialg_1(k7_freealg(A,B))
        & v6_unialg_1(k7_freealg(A,B))
        & v7_unialg_1(k7_freealg(A,B))
        & v8_unialg_1(k7_freealg(A,B))
        & v3_freealg(k7_freealg(A,B)) ) ) )).

fof(fc6_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B) )
     => ( ~ v3_struct_0(k15_freealg(A,B))
        & v3_unialg_1(k15_freealg(A,B))
        & v6_unialg_1(k15_freealg(A,B))
        & v7_unialg_1(k15_freealg(A,B))
        & v8_unialg_1(k15_freealg(A,B))
        & v3_freealg(k15_freealg(A,B)) ) ) )).

fof(rc6_freealg,axiom,(
    ? [A] :
      ( l1_unialg_1(A)
      & ~ v3_struct_0(A)
      & v3_unialg_1(A)
      & v6_unialg_1(A)
      & v7_unialg_1(A)
      & v8_unialg_1(A)
      & v2_unialg_2(A)
      & v3_freealg(A) ) )).

fof(d1_freealg,axiom,(
    ! [A] :
      ( v1_freealg(A)
    <=> r1_xboole_0(A,k5_numbers) ) )).

fof(d2_freealg,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v2_relat_1(A)
      <=> ~ r2_hidden(np__0,k2_relat_1(A)) ) ) )).

fof(d3_freealg,axiom,(
    $true )).

fof(d4_freealg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k4_finseq_1(u1_unialg_1(A)))
           => k1_freealg(A,B) = k1_funct_1(u1_unialg_1(A),B) ) ) ) )).

fof(d5_freealg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( m1_freealg(B,A)
          <=> u1_struct_0(k7_unialg_2(A,B)) = u1_struct_0(A) ) ) ) )).

fof(d6_freealg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_freealg(B,A)
         => ( v2_freealg(B,A)
          <=> ! [C] :
                ( ( ~ v3_struct_0(C)
                  & v6_unialg_1(C)
                  & v7_unialg_1(C)
                  & v8_unialg_1(C)
                  & l1_unialg_1(C) )
               => ( r1_unialg_2(A,C)
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,B,u1_struct_0(C))
                        & m2_relset_1(D,B,u1_struct_0(C)) )
                     => ? [E] :
                          ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(A),u1_struct_0(C))
                          & m2_relset_1(E,u1_struct_0(A),u1_struct_0(C))
                          & r1_alg_1(A,C,E)
                          & k2_partfun1(u1_struct_0(A),u1_struct_0(C),E,B) = D ) ) ) ) ) ) ) )).

fof(d7_freealg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ( v3_freealg(A)
      <=> ? [B] :
            ( m1_freealg(B,A)
            & v2_freealg(B,A) ) ) ) )).

fof(t1_freealg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_unialg_1(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( m1_freealg(B,A)
          <=> k7_unialg_2(A,B) = A ) ) ) )).

fof(d8_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B,C] :
          ( m2_relset_1(C,k2_xboole_0(k4_finseq_1(A),B),k3_finseq_2(k2_xboole_0(k4_finseq_1(A),B)))
         => ( C = k2_freealg(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k2_xboole_0(k4_finseq_1(A),B))
               => ! [E] :
                    ( m2_finseq_2(E,k2_xboole_0(k4_finseq_1(A),B),k3_finseq_2(k2_xboole_0(k4_finseq_1(A),B)))
                   => ( r2_hidden(k4_tarski(D,E),C)
                    <=> ( r2_hidden(D,k4_finseq_1(A))
                        & k1_funct_1(A,D) = k3_finseq_1(E) ) ) ) ) ) ) ) )).

fof(d9_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] : k3_freealg(A,B) = g1_lang1(k2_xboole_0(k4_finseq_1(A),B),k2_freealg(A,B)) ) )).

fof(t2_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( r1_tarski(k5_lang1(k3_freealg(A,B)),B)
          & k6_lang1(k3_freealg(A,B)) = k4_finseq_1(A) ) ) )).

fof(t3_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => k5_lang1(k3_freealg(A,B)) = B ) ) )).

fof(d10_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B,C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( r2_hidden(C,k4_finseq_1(A))
           => k4_freealg(A,B,C) = C ) ) ) )).

fof(d11_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_finseq_1(A,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_freealg(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(A))
               => ! [D] :
                    ( ( ~ v1_xboole_0(D)
                      & v1_funct_1(D)
                      & v1_unialg_1(D,k4_dtconstr(k3_freealg(A,B)))
                      & v2_unialg_1(D,k4_dtconstr(k3_freealg(A,B)))
                      & m2_relset_1(D,k3_finseq_2(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B))) )
                   => ( D = k5_freealg(A,B,C)
                    <=> ( k4_relset_1(k3_finseq_2(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B)),D) = k4_finseq_2(k4_finseq_4(k5_numbers,k5_numbers,A,C),k4_dtconstr(k3_freealg(A,B)))
                        & ! [E] :
                            ( m1_trees_4(E,k5_trees_3(u1_struct_0(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B)))
                           => ( r2_hidden(E,k4_relset_1(k3_finseq_2(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B)),D))
                             => k1_funct_1(D,E) = k12_trees_4(u1_struct_0(k3_freealg(A,B)),k4_freealg(A,B,C),E) ) ) ) ) ) ) ) ) ) )).

fof(d12_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_freealg(B) )
         => ! [C] :
              ( m2_finseq_1(C,k4_partfun1(k13_finseq_1(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B))))
             => ( C = k6_freealg(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(C))
                       => k1_funct_1(C,D) = k5_freealg(A,B,D) ) ) ) ) ) ) ) )).

fof(d13_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_freealg(B) )
         => k7_freealg(A,B) = g1_unialg_1(k4_dtconstr(k3_freealg(A,B)),k6_freealg(A,B)) ) ) )).

fof(t4_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_freealg(B) )
         => k3_unialg_1(k7_freealg(A,B)) = A ) ) )).

fof(t5_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_freealg(B) )
         => ~ v1_xboole_0(k8_freealg(A,B)) ) ) )).

fof(d15_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_freealg(B) )
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k3_freealg(A,B)))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k9_freealg(A,B),C)
                        & m2_relset_1(E,k9_freealg(A,B),C) )
                     => ( r2_hidden(D,k6_dtconstr(k3_freealg(A,B)))
                       => k10_freealg(A,B,C,D,E) = k1_funct_1(E,k2_trees_4(u1_struct_0(k3_freealg(A,B)),D)) ) ) ) ) ) ) )).

fof(d16_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B,C] :
          ( m1_subset_1(C,u1_struct_0(k3_freealg(A,B)))
         => ( ? [D] :
                ( v1_relat_1(D)
                & v1_funct_1(D)
                & v1_finseq_1(D)
                & r1_lang1(k3_freealg(A,B),C,D) )
           => k11_freealg(A,B,C) = C ) ) ) )).

fof(t6_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_freealg(B) )
         => v2_freealg(k9_freealg(A,B),k7_freealg(A,B)) ) ) )).

fof(d17_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m2_finseq_1(A,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(A))
               => ! [D] :
                    ( ( ~ v1_xboole_0(D)
                      & v1_funct_1(D)
                      & v1_unialg_1(D,k4_dtconstr(k3_freealg(A,B)))
                      & v2_unialg_1(D,k4_dtconstr(k3_freealg(A,B)))
                      & m2_relset_1(D,k3_finseq_2(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B))) )
                   => ( D = k13_freealg(A,B,C)
                    <=> ( k4_relset_1(k3_finseq_2(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B)),D) = k4_finseq_2(k4_finseq_4(k5_numbers,k5_numbers,A,C),k4_dtconstr(k3_freealg(A,B)))
                        & ! [E] :
                            ( m1_trees_4(E,k5_trees_3(u1_struct_0(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B)))
                           => ( r2_hidden(E,k4_relset_1(k3_finseq_2(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B)),D))
                             => k1_funct_1(D,E) = k12_trees_4(u1_struct_0(k3_freealg(A,B)),k4_freealg(A,B,C),E) ) ) ) ) ) ) ) ) ) )).

fof(d18_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => ! [C] :
              ( m2_finseq_1(C,k4_partfun1(k13_finseq_1(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B))))
             => ( C = k14_freealg(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(C))
                       => k1_funct_1(C,D) = k13_freealg(A,B,D) ) ) ) ) ) ) ) )).

fof(d19_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => k15_freealg(A,B) = g1_unialg_1(k4_dtconstr(k3_freealg(A,B)),k14_freealg(A,B)) ) ) )).

fof(t7_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => k3_unialg_1(k15_freealg(A,B)) = A ) ) )).

fof(t8_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => v2_unialg_2(k15_freealg(A,B)) ) ) )).

fof(t9_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => k6_unialg_2(k15_freealg(A,B)) != k1_xboole_0 ) ) )).

fof(d21_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k3_freealg(A,B)))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k17_freealg(A,B),C)
                        & m2_relset_1(E,k17_freealg(A,B),C) )
                     => ( r2_hidden(D,k5_lang1(k3_freealg(A,B)))
                       => k18_freealg(A,B,C,D,E) = k1_funct_1(E,k2_trees_4(u1_struct_0(k3_freealg(A,B)),D)) ) ) ) ) ) ) )).

fof(t10_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => v2_freealg(k17_freealg(A,B),k15_freealg(A,B)) ) ) )).

fof(dt_m1_freealg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ! [B] :
          ( m1_freealg(B,A)
         => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(existence_m1_freealg,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A) )
     => ? [B] : m1_freealg(B,A) ) )).

fof(dt_k1_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v6_unialg_1(A)
        & v7_unialg_1(A)
        & v8_unialg_1(A)
        & l1_unialg_1(A)
        & m1_subset_1(B,k5_numbers) )
     => m2_unialg_2(k1_freealg(A,B),u1_struct_0(A),k1_unialg_2(A)) ) )).

fof(dt_k2_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers) )
     => m2_relset_1(k2_freealg(A,B),k2_xboole_0(k4_finseq_1(A),B),k3_finseq_2(k2_xboole_0(k4_finseq_1(A),B))) ) )).

fof(dt_k3_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers) )
     => ( v1_lang1(k3_freealg(A,B))
        & l1_lang1(k3_freealg(A,B)) ) ) )).

fof(dt_k4_freealg,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k4_freealg(A,B,C),u1_struct_0(k3_freealg(A,B))) ) )).

fof(dt_k5_freealg,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B)
        & m1_subset_1(C,k5_numbers) )
     => ( ~ v1_xboole_0(k5_freealg(A,B,C))
        & v1_funct_1(k5_freealg(A,B,C))
        & v1_unialg_1(k5_freealg(A,B,C),k4_dtconstr(k3_freealg(A,B)))
        & v2_unialg_1(k5_freealg(A,B,C),k4_dtconstr(k3_freealg(A,B)))
        & m2_relset_1(k5_freealg(A,B,C),k3_finseq_2(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B))) ) ) )).

fof(dt_k6_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => m2_finseq_1(k6_freealg(A,B),k4_partfun1(k13_finseq_1(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B)))) ) )).

fof(dt_k7_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => ( ~ v3_struct_0(k7_freealg(A,B))
        & v3_unialg_1(k7_freealg(A,B))
        & v6_unialg_1(k7_freealg(A,B))
        & v7_unialg_1(k7_freealg(A,B))
        & v8_unialg_1(k7_freealg(A,B))
        & l1_unialg_1(k7_freealg(A,B)) ) ) )).

fof(dt_k8_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => m1_subset_1(k8_freealg(A,B),k1_zfmisc_1(u1_struct_0(k7_freealg(A,B)))) ) )).

fof(dt_k9_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => m1_freealg(k9_freealg(A,B),k7_freealg(A,B)) ) )).

fof(redefinition_k9_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => k9_freealg(A,B) = k8_freealg(A,B) ) )).

fof(dt_k10_freealg,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,u1_struct_0(k3_freealg(A,B)))
        & v1_funct_1(E)
        & v1_funct_2(E,k9_freealg(A,B),C)
        & m1_relset_1(E,k9_freealg(A,B),C) )
     => m1_subset_1(k10_freealg(A,B,C,D,E),C) ) )).

fof(dt_k11_freealg,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & m1_subset_1(C,u1_struct_0(k3_freealg(A,B))) )
     => m2_subset_1(k11_freealg(A,B,C),k1_numbers,k5_numbers) ) )).

fof(dt_k12_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => ( v2_freealg(k12_freealg(A,B),k7_freealg(A,B))
        & m1_freealg(k12_freealg(A,B),k7_freealg(A,B)) ) ) )).

fof(redefinition_k12_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v1_freealg(B) )
     => k12_freealg(A,B) = k8_freealg(A,B) ) )).

fof(dt_k13_freealg,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B)
        & m1_subset_1(C,k5_numbers) )
     => ( ~ v1_xboole_0(k13_freealg(A,B,C))
        & v1_funct_1(k13_freealg(A,B,C))
        & v1_unialg_1(k13_freealg(A,B,C),k4_dtconstr(k3_freealg(A,B)))
        & v2_unialg_1(k13_freealg(A,B,C),k4_dtconstr(k3_freealg(A,B)))
        & m2_relset_1(k13_freealg(A,B,C),k3_finseq_2(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B))) ) ) )).

fof(dt_k14_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B) )
     => m2_finseq_1(k14_freealg(A,B),k4_partfun1(k13_finseq_1(k4_dtconstr(k3_freealg(A,B))),k4_dtconstr(k3_freealg(A,B)))) ) )).

fof(dt_k15_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B) )
     => ( ~ v3_struct_0(k15_freealg(A,B))
        & v3_unialg_1(k15_freealg(A,B))
        & v6_unialg_1(k15_freealg(A,B))
        & v7_unialg_1(k15_freealg(A,B))
        & v8_unialg_1(k15_freealg(A,B))
        & l1_unialg_1(k15_freealg(A,B)) ) ) )).

fof(dt_k16_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B) )
     => m1_subset_1(k16_freealg(A,B),k1_zfmisc_1(u1_struct_0(k15_freealg(A,B)))) ) )).

fof(dt_k17_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B) )
     => m1_freealg(k17_freealg(A,B),k15_freealg(A,B)) ) )).

fof(redefinition_k17_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B) )
     => k17_freealg(A,B) = k16_freealg(A,B) ) )).

fof(dt_k18_freealg,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,u1_struct_0(k3_freealg(A,B)))
        & v1_funct_1(E)
        & v1_funct_2(E,k17_freealg(A,B),C)
        & m1_relset_1(E,k17_freealg(A,B),C) )
     => m1_subset_1(k18_freealg(A,B,C,D,E),C) ) )).

fof(dt_k19_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B) )
     => ( v2_freealg(k19_freealg(A,B),k15_freealg(A,B))
        & m1_freealg(k19_freealg(A,B),k15_freealg(A,B)) ) ) )).

fof(redefinition_k19_freealg,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_finseq_1(A,k5_numbers)
        & v1_freealg(B) )
     => k19_freealg(A,B) = k16_freealg(A,B) ) )).

fof(d14_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_freealg(B) )
         => k8_freealg(A,B) = a_2_0_freealg(A,B) ) ) )).

fof(d20_freealg,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v2_relat_1(A)
        & m1_trees_4(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_freealg(B)
         => k16_freealg(A,B) = a_2_1_freealg(A,B) ) ) )).

fof(fraenkel_a_2_0_freealg,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_trees_4(B,k1_numbers,k5_numbers)
        & ~ v1_xboole_0(C)
        & v1_freealg(C) )
     => ( r2_hidden(A,a_2_0_freealg(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(k3_freealg(B,C)))
            & A = k2_trees_4(u1_struct_0(k3_freealg(B,C)),D)
            & r2_hidden(D,k6_dtconstr(k3_freealg(B,C))) ) ) ) )).

fof(fraenkel_a_2_1_freealg,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & ~ v2_relat_1(B)
        & m1_trees_4(B,k1_numbers,k5_numbers)
        & v1_freealg(C) )
     => ( r2_hidden(A,a_2_1_freealg(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(k3_freealg(B,C)))
            & A = k2_trees_4(u1_struct_0(k3_freealg(B,C)),D)
            & r2_hidden(D,k5_lang1(k3_freealg(B,C))) ) ) ) )).
%------------------------------------------------------------------------------